Research Article A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media Qian Zhang 1,2 and Chao Huang 1,3 College of Mathematics and Statistics, Shenzhen University, China Shenzhen Key Laboratory of Advanced Machine Learning and Applications, China Guangdong Key Laboratory of Intelligent Information Processing, China Correspondence should be addressed to Qian Zhang; [email protected] Received 25 March 2019; Accepted 15 May 2019; Published 12 June 2019 Academic Editor: Tepper L. Gill Copyright © 2019 Qian Zhang and Chao Huang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we present an uncoupled leap-frog finite difference method for the system of equations arising from sweat transport through porous textile media. Based on physical mechanisms, the sweat transport can be viewed as the multicomponent flow that coupled the heat and moisture transfer, such that the system is nonlinear and strongly coupled. e leap-frog method is proposed to solve this system, with the second order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of the solution to the system with optimal error estimates in the discrete 2 norm. Numerical simulations are presented and analyzed, respectively. 1. Introduction Single/multicomponent flow in porous textile media at- tracted considerable attention in the last several decades. See [1–4] for the single-component models and [5–9] for the multicomponent models. In this paper, we study the multicomponent sweat transport coupled with vapor and heat in porous textile media. In [10], Ye et al. proposed a quasi-steady-state single-component model which consists of a steady-state air equation and dynamic state equations for other components. Under certain conditions, the multicom- ponent model reduces to a new single-component model, and the physical process can be viewed as sweat transport (vapor and heat flow) governed by the conservation of mass and energy: () + ( ) = −Γ ce , (1) ( V ) + ( ) − ( ) = Γ ce , (2) where is the porosity of the media, is the vapor concen- tration, is the temperature, is the thermal conductivity, is the latent heat of evaporation/condensation, and is the molecular weight of water. e effective volumetric heat capacity V is defined by V = + (1 − ) V , (3) where is the molar heat capacity and V is the volumet- ric heat capacity of fiber. By Darcy’s law, the gas velocity is defined as =− , (4) where is the permeability and is the dynamic viscosity, which usually is density-dependent for the compressible flow. Here we choose a linear form of fl ], where ] is a certain constant. By the Hertz-Knudsen equation [11], the phase change rate Γ is defined as Γ ce = Γ ( − sat () √ ), (5) Hindawi Journal of Mathematics Volume 2019, Article ID 8649308, 16 pages https://doi.org/10.1155/2019/8649308
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Research ArticleA Leap-Frog Finite Difference Method for Strongly CoupledSystem from Sweat Transport in Porous Textile Media
Qian Zhang 12 and Chao Huang13
1College of Mathematics and Statistics Shenzhen University China2Shenzhen Key Laboratory of Advanced Machine Learning and Applications China3Guangdong Key Laboratory of Intelligent Information Processing China
Correspondence should be addressed to Qian Zhang mazhangqszueducn
Received 25 March 2019 Accepted 15 May 2019 Published 12 June 2019
Academic Editor Tepper L Gill
Copyright copy 2019 Qian Zhang andChaoHuangThis is an open access article distributed under the Creative Commons AttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly cited
In this paper we present an uncoupled leap-frog finite difference method for the system of equations arising from sweat transportthrough porous textile media Based on physical mechanisms the sweat transport can be viewed as the multicomponent flow thatcoupled the heat andmoisture transfer such that the system is nonlinear and strongly coupledThe leap-frogmethod is proposed tosolve this system with the second order accuracy in both spatial and temporal directions We prove the existence and uniqueness ofthe solution to the system with optimal error estimates in the discrete 1198712 norm Numerical simulations are presented and analyzedrespectively
1 Introduction
Singlemulticomponent flow in porous textile media at-tracted considerable attention in the last several decadesSee [1ndash4] for the single-component models and [5ndash9] forthe multicomponent models In this paper we study themulticomponent sweat transport coupled with vapor andheat in porous textile media In [10] Ye et al proposed aquasi-steady-state single-componentmodel which consists ofa steady-state air equation and dynamic state equations forother components Under certain conditions the multicom-ponentmodel reduces to a new single-componentmodel andthe physical process can be viewed as sweat transport (vaporand heat flow) governed by the conservation of mass andenergy 120597120597119905 (120598119862) + 120597120597119909 (119906119892120598119862) = minusΓce (1)
120597120597119905 (119862V119905119879) + 120597120597119909 (120598119906119892119862119898119892119862119879) minus 120597120597119909 (120581120597119879120597119909)= 120582119872119908Γce
(2)
where 120598 is the porosity of the media 119862 is the vapor concen-tration 119879 is the temperature 120581 is the thermal conductivity120582 is the latent heat of evaporationcondensation and 119872119908 isthe molecular weight of water The effective volumetric heatcapacity 119862V119905 is defined by119862V119905 = 120598119862119898119892119862 + (1 minus 120598) 119862V119891 (3)
where 119862119898119892 is the molar heat capacity and 119862V119891 is the volumet-ric heat capacity of fiber
By Darcyrsquos law the gas velocity 119906119892 is defined as
119906119892 = minus119896120583 120597119875120597119909 (4)
where 119896 is the permeability and 120583 is the dynamic viscositywhich usually is density-dependent for the compressible flowHere we choose a linear form of 120583 fl ]119862 where ] is a certainconstant
By the Hertz-Knudsen equation [11] the phase changerate Γ119888119890 is defined as
Γce = 120573Γ (119875 minus 119875sat (119879)radic119879 ) (5)
HindawiJournal of MathematicsVolume 2019 Article ID 8649308 16 pageshttpsdoiorg10115520198649308
2 Journal of Mathematics
where 120573Γ is a positive constant the saturation pressure 119875sat isdetermined from experimental measurements [12] and thepressure 119875 is given by 119875 = 119877119862119879 where 119877 is the universal gasconstant
With nondimensionalization the sweat transport process(1)-(2) can be described by the following system
119888119905 minus ((119888120579)119909)119909 = minusΓ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (6)
(119888120579)119905 + (120590120579)119905 minus ((119888120579)119909 120579)119909 minus 120581120579119909119909 = 120582Γ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (7)
where (sdot)119909 = 120597120597119909 (sdot)119905 = 120597120597119905 Γ(119888 120579) = 119888radic120579 minus 119901119904(120579)and 119901119904(120579) sim 119875119904119886119905(120579)radic120579 is a smooth and increasing functionsatisfying 119901s(0) = 0
Since the right boundary is exposed to environment andthe left boundary is connected to the body we considercommonly used Robin type boundary conditions
(119888120579)119909 = 1205721 (119888 minus 1205831) 119909 = 0 0 lt 119905 le 119879 (8)
(119888120579)119909 = 1205722 (1205832 minus 119888) 119909 = 119871 0 lt 119905 le 119879 (9)
120579119909 = 1205731 (120579 minus ]1) 119909 = 0 0 lt 119905 le 119879 (10)
120579119909 = 1205732 (]2 minus 120579) 119909 = 119871 0 lt 119905 le 119879 (11)
and the initial conditions
119888 (119909 0) = 1198880 (119909) 120579 (119909 0) = 1205790 (119909) 0 le 119909 le 119871(12)
Physically parameters 120572119894 120573119894 120583119894 ]119894 119894 = 1 2 and 120590 arenonnegative constants [1 2 6] We define initial conditionparameters 1198880(119909) ge 119888 1205790(119909) ge 120579 with 119888 and 120579 being positiveconstants
Due to the strong nonlinearity and the coupling of thesystem both theoretical and numerical analyses of the systemare difficult Numerical analysis for some related systemsof parabolicelliptic equations can be found in [13ndash20]Existence and uniqueness of a classical solution for a steady-state model was given in [10] Existence of a weak solutionfor the corresponding dynamic models was given in [21 22]Positivity of temperature and nonnegativity of vapor densitywere also proved here Recently a finite difference methodsecond-order in space and first-order in time for the system(6)-(12) was presented in [23] where the backward semi-implicit Euler scheme is applied in the temporal direction andcentral finite difference approximations are used in the spatialdirection In [23] authors presented optimal error estimatesunder the assumption that the step size 120591 and ℎ are smallerthan a positive constant
In this paper we propose an uncoupled leap-frog finitedifference method for the system (6)-(12) with second-orderaccuracy in both spatial and temporal directions We provethe existence and uniqueness of a solution to the finite
difference system with optimal error estimates in the discrete1198712 norm under the condition that the mesh size 120591 and ℎ aresmaller than a positive constant which depends solely uponthe physical parameters involved in the equations Due tothe strong nonlinearity and the coupling of equations themethod presented in [23] does not apply to the leap-frogscheme directly One of the difficulties is to show convergenceof the numerical solutionwithout restriction on the grid ratioIn this paper we assume that the solution (119888(119909 119905) 120579(119909 119905)) tothe system (6)-(12) satisfies that
119888min le 119888 (119909 119905) le 119888max120579min le 120579 (119909 119905) le 120579max(13)
for some positive constants 119888min 119888max 120579max and 120579minThe manuscript is organized as follows in Section 2
we present an uncoupled leap-frog finite difference methodfor the nonlinear sweat transport system In Section 3 weprove the existence and uniqueness of the solution to thesweat transport system with the optimal error estimate inthe discrete 1198712 norm Numerical results will be presented inSection 4 to support our theoretical results
2 The Leap-Frog Finite Difference Scheme
For convenience of calculations we add the equation (6)times 120579 into the equation (7) thus the governing system (6)-(7) can be modified as
119888119905 minus ((119888120579)119909)119909 = minusΓ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (14)
(119888 + 120590) 120579119905 minus (119888120579)119909 120579119909 minus 120581120579119909119909 = (120582 + 120579) Γ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (15)
Due to the practical interest in a long time period say 8ndash24hours we present an uncoupled leap-frog finite differencescheme in the temporal direction and the central finitedifference (volume) scheme in the spatial direction for theabove system with the initialboundary conditions (8)-(12)
Let T be a positive number let Ωℎ = 119909119894 | 119909119894 =119894ℎ 0 le 119894 le 119872 be a uniform partition in [0 119871] and letΩ120591 = 119905119899 | 119905119899 = 119899120591 0 le 119899 le 119873 be a uniform partitionin [0T] where ℎ = 119871119872 and 120591 = T119873 are the step sizein the spatial and temporal directions respectively Denote119909119894+12 = (119909119894 + 119909119894+1)2 in the spatial cell and let V119899119894 | 0 le 119894 le119872 0 le 119899 le 119873 be a mesh function defined on Ωℎ120591 whereΩℎ120591 = Ωℎ times Ω120591 Some notations are introduced below
nabla119905V119899119894 = 12120591 (V119899+1
119894 minus V119899minus1119894 )
V119899119894 = 12 (V119899+1119894 + V119899minus1
119894 ) V119899119894+12 = 12 (V119899+1
119894+1 + V119899minus1119894 )
V119899119894+12 = 12 (V119899119894 + V119899119894+1)
Journal of Mathematics 3
120575119909V119899119894+12 = 1ℎ (V119899119894+1 minus V119899119894 ) 120575lowast119909V
119899119894 = 1ℎ (V119899119894+12 minus V119899119894minus12)
120575119909V119899119894+12 = 12 (120575119909V
119899+1119894+12 + 120575119909V
119899minus1119894+12)
1205752119909V
119899119894+12 = 1ℎ (120575119909V
119899119894+12 minus 120575119909V
119899119894minus12)
(16)
from which
(120575119909V119898119894+12) V119899119894+12 + V119898119894+12120575119909V
119899119894+12 = 120575119909 (V119898119894+12V
119899119894+12) (17)
The discrete system is defined by
nabla119905119888119899119894 minus 1ℎ [(119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12)minus (119888119899119894minus12120575119909120579119899
119894minus12 + 120579119899119894minus12120575119909119888119899119894minus12)] = minusΓ (119888119899119894 120579119899
119894 ) 1 le 119894 le 119872 minus 1
(18)
nabla1199051198881198990 minus 2ℎ [(1198881198991212057511990912057911989912 + 120579119899
1212057511990911988811989912) minus 1205721 (1198881198990 minus 1205831)]= minusΓ (1198881198990 120579119899
0) (19)
nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872)minus (119888119899119872minus12120575119909120579119899
119872minus12 + 120579119899119872minus12120575119909119888119899119872minus12)]
= minusΓ (119888119899119872 120579119899119872)
(20)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581ℎ (120575119909120579119899119894+12 minus 120575119909120579119899
119894minus12) = (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) 1 le 119894 le 119872 minus 1
(21)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [120575119909120579119899
12
minus 1205731 (1205791198990 minus ]1)] = (120582 + 120579119899
0) Γ (1198881198990 1205791198990)
(22)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 120575119909120579119899
119872minus12] = (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872) 1 le 119899 le 119873 minus 1
(23)
and the discrete initial conditions
1198880119894 = 1198880 (119909119894) 1205790119894 = 1205790 (119909119894) (24)
1198881119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) 1205791119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) 0 le 119894 le 119872 (25)
where
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12 (26)
The computational procedure of the uncoupled leap-frogscheme at each time step is listed below
Step 1 The vapor concentration 119888119899+1119895 can be calculated by
solving the tridiagonal linear systems defined in (18)-(20)
Step 2 With the updated vapor concentration 119888119899+1119895 we can
get 1198881198990 and 119906119899119894+12 correspondingly
Step 3 Finally the temperature 120579119899+1119895 can be obtained by
solving the tridiagonal linear system (21)-(23)
3 The Leap-Frog Scheme andthe Optimal Error Estimate
In this section we will show the existence and uniquenessof the solution to the system (18)-(26) with optimal errorestimates in the discrete 1198712 norm Let V = V119899119895 119872119895=0 and 119911 =119911119895119872119895=0 be two mesh functions on Ωℎ We define the innerproduct and norms by
(V 119911) = ℎ(12V01199110 +119872minus1sum119894=1
V119894119911119894 + 12V119872119911119872) V = radicℎ[12 (V0)2 +
119872minus1sum119894=1
(V119894)2 + 12 (V119872)2]Vinfin = max
0le119894le119872
1003816100381610038161003816V1198941003816100381610038161003816 1003817100381710038171003817120575119909V
1003817100381710038171003817 = radicℎ119872minus1sum119894=0
(V119894+1 minus V119894ℎ )2100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 = radicℎ119872minus1sum
119894=1
(1205752119909V119894)2
(27)
Let (119862 Θ) be the solution of the system (6)-(12) and 119862119899119894 =119888(119909119894 119905119899) Θ119899
119894 = 120579(119909119894 119905119899) The error functions are defined by
119888119899119894 = 119862119899119894 minus 119888119899119894
120579119899119894 = Θ119899
119894 minus 120579119899119894 0 le 119894 le 119872 0 le 119899 le 119873
(28)
We state our main result in the theorem below
Theorem 1 Suppose that the solution (119888 V) of the system (6)-(12) is in 11986243([0 119871] times [0T]) satisfying (13) en there existpositive constants ℎ0 and 1198640 independent of ℎ and 120591 such that
4 Journal of Mathematics
when 120591 le 119864119888ℎ le ℎ0 the finite difference scheme (18)-(26) isuniquely solvable and
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 119899sum119898=1
(10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172)le 1198640 (1205912 + ℎ2)2 1 le 119899 le 119873 minus 1
(29)
To prove the theorem we make a stronger assumptionthat there exists 1198640 gt 0 independent of 119899 ℎ 120591 such that theinequality
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (30)
holds for 119899 le 119896minus1We prove the assumption and the theoremby inductionmethod By the initial condition (26) this is truefor 119899 = 0 In the next subsection we will show that this is alsotrue for 119899 = 1 In this part we let 119864119888 be a generic positiveconstant which is associated with the physical parameters 120590120582 120581 119888min 119888max 120579min 120579max the parameters involved in initialand boundary conditions and the solution of the system (6)-(12) 119864119888 is independent of time step 119899 mesh size ℎ 120591 andconstant 1198640
31 e Leap-Frog Scheme and Preliminaries For conve-nience of calculations we further introduce some notationsLet 119906 = (119888120579)119909 119908 = 120579119909 thus the sweat transport system (6)-(7)can be reduced to
119888119905 minus 119906119909 = minusΓ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (31)
119906 = (119888120579119909 + 120579119888119909) 0 le 119909 le 119871 0 lt 119905 le T (32)
(119888 + 120590) 120579119905 minus 119906120579119909 minus 120581119908119909 = (120582 + 120579) Γ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (33)
119908 = 120579119909 0 le 119909 le 119871 0 lt 119905 le T (34)
with the initial and boundary conditions
119906 = 1205721 (119888 minus 1205831) 119909 = 0 0 lt 119905 le T119906 = 1205722 (1205832 minus 119888) 119909 = 119871 0 lt 119905 le T119908 = 1205731 (120579 minus ]1) 119909 = 0 0 lt 119905 le T119908 = 1205732 (]2 minus 120579) 119909 = 119871 0 lt 119905 le T
119888 (119909 0) = 1198880 (119909) 120579 (119909 0) = 1205790 (119909) 0 le 119909 le 119871
(35)
The discrete leap-frog system (18)-(23) is modified as
nabla119905119888119899119894 minus 120575lowast119909119906119899
119894 = minusΓ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
nabla1199051198881198990 minus 2ℎ [11990611989912 minus 1205721 (1198881198990 minus 1205831)] = minusΓ (1198881198990 120579119899
0) nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872) minus 119906119899
119872minus12] = minusΓ (119888119899119872 120579119899119872)
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12
0 le 119894 le 119872 minus 1(119888119899119894 + 120590)nabla119905120579119899
119894 minus 12 (119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12)minus 120581120575lowast
119909119908119899119894 = (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912
minus 2120581ℎ [11990811989912 minus 1205731 (120579119899
0 minus ]1)] = (120582 + 1205791198990) Γ (1198881198990 120579119899
0) (119888119899119872 + 120590)nabla119905120579119899
119872 minus 119906119899119872minus12120575119909120579119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 119908119899
119872minus12]= (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)
119908119899119894+12 = 120575119909120579119899
119894+12 0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(36)
Let 119880119899119894 = 119906(119909119894 119905119899) = (119888120579)119909(119909119894 119905119899) and 119882119899
119894 = 119908(119909119894 119905119899) =120579119909(119909119894 119905119899) We denote by 119906119899119894 and 119908119899
119894 the corresponding finitedifference solution and
119899119894+12 = 119880119899
119894+12 minus 119906119899119894+12
119908119899119894+12 = 119882119899
119894+12 minus 119908119899119894+12
0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1(37)
We get
nabla119905119862119899119894 minus 120575lowast
119909119880119899119894 = minusΓ (119862119899
119894 Θ119899119894 ) + 119877119899
119888119894 1 le 119894 le 119872 minus 1nabla119905119862119899
0 minus 2ℎ [11988011989912 minus 1205721 (119862119899
0 minus 1205831)] = minusΓ (1198621198990 Θ119899
0) + 1198771198991198880
nabla119905119862119899119872 minus 2ℎ [1205722 (1205832 minus 119862119899
119872) minus 119880119899119872minus12] = minusΓ (119862119899
119872 Θ119899119872)
+ 119877119899119888119872
119880119899119894+12 = 119862119899
119894+12120575119909Θ119899119894+12 + Θ119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+120 le 119894 le 119872 minus 1
(119862119899119894 + 120590)nabla119905Θ119899
119894 minus 12 (119880119899119894minus12120575119909Θ119899
119894minus12 + 119880119899119894+12120575119909Θ119899
119894+12)minus 120581120575lowast
119909119882119899119894 = (120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) + 1198771198991205791198941 le 119894 le 119872 minus 1
Journal of Mathematics 5
(1198621198990 + 120590)nabla119905Θ119899
0 minus 11988011989912120575119909Θ119899
12
minus 2120581ℎ [11988211989912 minus 1205731 (Θ119899
0 minus ]1)]= (120582 + Θ119899
0) Γ (1198621198990 Θ119899
0) + 1198771198991205790
(119862119899119872 + 120590)nabla119905Θ119899
119872 minus 119880119899119872minus12120575119909Θ119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus Θ119899119872) minus119882119899
119872minus12]= (120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) + 119877119899120579119872
119882119899119894+12 = 120575119909Θ119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(38)
and the initial conditions
1198620119894 = 1198880 (119909119894)
Θ0119894 = 1205790 (119909119894)
1198621119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) + 1205911198771
119888119894Θ1
119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) + 1205911198771120579119894
(39)
where 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 le 1198641 (120591 + ℎ2) 0 le 119894 le 11987210038161003816100381610038161003816119877119899119888119894
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119894
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 1 le 119894 le 119872 minus 110038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119908119894+12
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 0 le 119894 le 119872 minus 1100381610038161003816100381610038161198771198991198880
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119888119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 100381610038161003816100381610038161198771198991205790
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ) 1 le 119899 le 119873 minus 1
(40)
Subtracting the system (36) from the system (38) we get theerror equations
nabla119905119888119899119894 minus 120575lowast119909 119899
119894 = minus [Γ (119862119899119894 Θ119899
119894 ) minus Γ (119888119899119894 120579119899119894 )] + 119877119899
119888119894
fl 119877119899
119888119894 1 le 119894 le 119872 minus 1 (41)
nabla1199051198881198990 minus 2ℎ [11989912 minus 12057211198881198990 ] = minus [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)]+ 119877119899
1198880 fl 119877119899
1198880(42)
nabla119905119888119899119872 minus 2ℎ [minus1205722119888119899119872 minus 119899119872minus12]
= minus [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] + 119877119899
119888119872 fl 119877119899
119888119872(43)
119899119894+12 = 119888119899119894+12120575119909Θ119899
119894+12 + 120579119899119894+12120575119909119862119899
119894+12
+ 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12 + 119877119899119906119894+120 le 119894 le 119872 minus 1
(44)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581120575lowast119909119908119899
119894 = minus119888119899119894 nabla119905Θ119899119894
+ 12 (119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)+ [(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )]+ 119877119899
120579119894 fl 119877119899
120579119894 1 le 119894 le 119872 minus 1
(45)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [119908119899
12 minus 12057311205791198990]
= minus1198881198990nabla119905Θ1198990 + 119899
12120575119909Θ11989912
+ [(120582 + Θ1198990) Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)]
+ 1198771198991205790 fl 119877119899
1205790(46)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [minus1205732120579119899119872 minus 119908119899
119872minus12] = minus119888119899119872nabla119905Θ119899119872
+ 119899119872minus12120575119909Θ119899
119872minus12
+ [(120582 + Θ119899119872) Γ (119862119899
119872 Θ119899119872) minus (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)]
+ 119877119899120579119872 fl 119877119899
120579119872
(47)
119908119899119894+12 = 120575119909120579119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1 (48)
and
1198880119894 = 01205790119894 = 01198881119894 = 1205911198771
1198881198941205791119894 = 1205911198771
120579119894(49)
and by (40) we can directly derive the inequality
100381710038171003817100381710038171198881100381710038171003817100381710038172 + 100381710038171003817100381710038171205791100381710038171003817100381710038172 + 100381710038171003817100381710038171205751199091205791100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (50)
To prove our main theorem the following formula will beoften used
[V121199110 + ℎ119872minus1sum119894=1
120575119909V119894119911119894 minus V119872minus12119911119872]= 119872minus1sum
119894=0
V119894+12 (119911119896119894 minus 119911119896
119894+1) (51)
In the following lemma we present discrete Sobolev interpo-lation formulas and the proof can be found in [24]
6 Journal of Mathematics
Lemma 2 Let V and 119911 be two mesh functions en for anypositive constant 120598
V2infin le 120598 1003817100381710038171003817120575119909V10038171003817100381710038172 + (1120598 + 1119871) V2 (52)
1003817100381710038171003817120575119909V1003817100381710038171003817infin le 120598 100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 + 119864119888
1003817100381710038171003817120575119909V10038171003817100381710038172 (53)
Lemma 3
10038171003817100381710038171198881198991003817100381710038171003817infin 1003817100381710038171003817100381712057911989910038171003817100381710038171003817infin le 3119864120 (12059174 + ℎ32) 1 le 119899 le 119896 (54)1003817100381710038171003817100381711988811989911990910038171003817100381710038171003817 10038171003817100381710038171003817120579119899
119909
10038171003817100381710038171003817 le 119864120 (12059132 + ℎ) 1 le 119899 le 119896 minus 1 (55)
Proof From (30) for 0 le 119899 le 119896 minus 1 we have10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (56)
When 120591 le ℎ with the inverse inequality we have
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infin le 2ℎminus1 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 le 2ℎminus11198640 (1205912 + ℎ2)2le 81198640ℎ3 (57)
When ℎ le 120591 by taking 120598 = 12059112 in Lemma 2
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infinle 12059112 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 + (120591minus12 + 119871minus1) 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172le (120591minus12 + 12059112119871minus1) 1198640 (1205912 + ℎ2)2 le 8119864012059172
1 le 119894 le 119872 minus 1 0 le 119899 le 119896 minus 1(58)
The first part of (54) is obtained and the second part and theinequality (55) can be proved similarly
In addition by Lemma 3 there exist constants 1198643 gt 0 and1199040 gt 0 such that when ℎ 120591 le 1199040119888min2 le 119888119899119894 le 2119888max120579min2 le 120579119899
119894 le 2120579max0 le 119894 le 119872 minus 1 1 le 119899 le 119896
(59)
and 1003816100381610038161003816nabla119905Θ1198991198941003816100381610038161003816 1003816100381610038161003816nabla119905119862119899
1198941003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816120575119909119862119899
119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119862119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 le 11986430 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816119888119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120579119899119894+12
10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 119896
10038161003816100381610038161003816Γ (119888119899119894+12 120579119899119894+12)10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 1198961003816100381610038161003816(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899
119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 1198961003816100381610038161003816Γ (119862119899
119894 Θ119899119894 ) minus Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816 le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 119896(60)
32 e Existence and Uniqueness Since the coefficientmatrix in the system (18)-(20) is strictly diagonally dominantthus the system (18)-(20) has a unique solution 119888119896+1
119894 Here wewill discuss the boundedness of 119888119896+1
119894 Multiplying (41)-(43) by ℎ119888119899119894 ℎ1198881198990 2 and ℎ1198881198991198722 respec-
tively we get
(nabla119905119888119899 119888119899) + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162 = [119899121198881198990
+ ℎ119872minus1sum119894=1
120575lowast119909 119899
119894 119888119899119894 + 119899119872minus12119888119899119872] + ℎ119872minus1sum
119894=1
119877119899119888119894119888119899119894 + ℎ2
sdot 11987711989911988801198881198990 + ℎ2119877119899
119888119872119888119899119872minus ℎ12 [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)] 1198881198990+ 119872minus1sum
119894=1
[Γ (119862119899119894 Θn
119894 ) minus Γ (119888119899119894 120579119899119894 )] 119888119899119894
+ 12 [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] 119888119899119872 fl 1198691 + 1198692
+ 1198693 1 le 119899 le 119896
(61)
With (44) (51) (59) (60) and (40) we see that
minus 1198691 = minus119899121198881198990 minus ℎ119872minus1sum
119894=1
120575119909119899119894 119888119899119894 + 119899
Mminus12119888119899119872= ℎ119872minus1sum
119894=0
119899119894+12120575119909119888119899119894+12 = ℎ119872minus1sum
119894=0
[119888119899119894+12120575119909Θ119899119894+12
+ 120579119899119894+12120575119909119862119899
119894+12 + 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119877119899119906119894+12] 120575119909119888119899119894+12 ge 120579min2 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 2119888max
1003817100381710038171003817100381712057511990912057911989910038171003817100381710038171003817sdot 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817 minus 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817minus ℎ119872minus1sum
119894=0
10038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119888119899119894+12
10038161003816100381610038161003816 ge 120579min4 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172minus 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(62)
Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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2 Journal of Mathematics
where 120573Γ is a positive constant the saturation pressure 119875sat isdetermined from experimental measurements [12] and thepressure 119875 is given by 119875 = 119877119862119879 where 119877 is the universal gasconstant
With nondimensionalization the sweat transport process(1)-(2) can be described by the following system
119888119905 minus ((119888120579)119909)119909 = minusΓ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (6)
(119888120579)119905 + (120590120579)119905 minus ((119888120579)119909 120579)119909 minus 120581120579119909119909 = 120582Γ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (7)
where (sdot)119909 = 120597120597119909 (sdot)119905 = 120597120597119905 Γ(119888 120579) = 119888radic120579 minus 119901119904(120579)and 119901119904(120579) sim 119875119904119886119905(120579)radic120579 is a smooth and increasing functionsatisfying 119901s(0) = 0
Since the right boundary is exposed to environment andthe left boundary is connected to the body we considercommonly used Robin type boundary conditions
(119888120579)119909 = 1205721 (119888 minus 1205831) 119909 = 0 0 lt 119905 le 119879 (8)
(119888120579)119909 = 1205722 (1205832 minus 119888) 119909 = 119871 0 lt 119905 le 119879 (9)
120579119909 = 1205731 (120579 minus ]1) 119909 = 0 0 lt 119905 le 119879 (10)
120579119909 = 1205732 (]2 minus 120579) 119909 = 119871 0 lt 119905 le 119879 (11)
and the initial conditions
119888 (119909 0) = 1198880 (119909) 120579 (119909 0) = 1205790 (119909) 0 le 119909 le 119871(12)
Physically parameters 120572119894 120573119894 120583119894 ]119894 119894 = 1 2 and 120590 arenonnegative constants [1 2 6] We define initial conditionparameters 1198880(119909) ge 119888 1205790(119909) ge 120579 with 119888 and 120579 being positiveconstants
Due to the strong nonlinearity and the coupling of thesystem both theoretical and numerical analyses of the systemare difficult Numerical analysis for some related systemsof parabolicelliptic equations can be found in [13ndash20]Existence and uniqueness of a classical solution for a steady-state model was given in [10] Existence of a weak solutionfor the corresponding dynamic models was given in [21 22]Positivity of temperature and nonnegativity of vapor densitywere also proved here Recently a finite difference methodsecond-order in space and first-order in time for the system(6)-(12) was presented in [23] where the backward semi-implicit Euler scheme is applied in the temporal direction andcentral finite difference approximations are used in the spatialdirection In [23] authors presented optimal error estimatesunder the assumption that the step size 120591 and ℎ are smallerthan a positive constant
In this paper we propose an uncoupled leap-frog finitedifference method for the system (6)-(12) with second-orderaccuracy in both spatial and temporal directions We provethe existence and uniqueness of a solution to the finite
difference system with optimal error estimates in the discrete1198712 norm under the condition that the mesh size 120591 and ℎ aresmaller than a positive constant which depends solely uponthe physical parameters involved in the equations Due tothe strong nonlinearity and the coupling of equations themethod presented in [23] does not apply to the leap-frogscheme directly One of the difficulties is to show convergenceof the numerical solutionwithout restriction on the grid ratioIn this paper we assume that the solution (119888(119909 119905) 120579(119909 119905)) tothe system (6)-(12) satisfies that
119888min le 119888 (119909 119905) le 119888max120579min le 120579 (119909 119905) le 120579max(13)
for some positive constants 119888min 119888max 120579max and 120579minThe manuscript is organized as follows in Section 2
we present an uncoupled leap-frog finite difference methodfor the nonlinear sweat transport system In Section 3 weprove the existence and uniqueness of the solution to thesweat transport system with the optimal error estimate inthe discrete 1198712 norm Numerical results will be presented inSection 4 to support our theoretical results
2 The Leap-Frog Finite Difference Scheme
For convenience of calculations we add the equation (6)times 120579 into the equation (7) thus the governing system (6)-(7) can be modified as
119888119905 minus ((119888120579)119909)119909 = minusΓ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (14)
(119888 + 120590) 120579119905 minus (119888120579)119909 120579119909 minus 120581120579119909119909 = (120582 + 120579) Γ (119888 120579) 0 lt 119909 lt 119871 119905 gt 0 (15)
Due to the practical interest in a long time period say 8ndash24hours we present an uncoupled leap-frog finite differencescheme in the temporal direction and the central finitedifference (volume) scheme in the spatial direction for theabove system with the initialboundary conditions (8)-(12)
Let T be a positive number let Ωℎ = 119909119894 | 119909119894 =119894ℎ 0 le 119894 le 119872 be a uniform partition in [0 119871] and letΩ120591 = 119905119899 | 119905119899 = 119899120591 0 le 119899 le 119873 be a uniform partitionin [0T] where ℎ = 119871119872 and 120591 = T119873 are the step sizein the spatial and temporal directions respectively Denote119909119894+12 = (119909119894 + 119909119894+1)2 in the spatial cell and let V119899119894 | 0 le 119894 le119872 0 le 119899 le 119873 be a mesh function defined on Ωℎ120591 whereΩℎ120591 = Ωℎ times Ω120591 Some notations are introduced below
nabla119905V119899119894 = 12120591 (V119899+1
119894 minus V119899minus1119894 )
V119899119894 = 12 (V119899+1119894 + V119899minus1
119894 ) V119899119894+12 = 12 (V119899+1
119894+1 + V119899minus1119894 )
V119899119894+12 = 12 (V119899119894 + V119899119894+1)
Journal of Mathematics 3
120575119909V119899119894+12 = 1ℎ (V119899119894+1 minus V119899119894 ) 120575lowast119909V
119899119894 = 1ℎ (V119899119894+12 minus V119899119894minus12)
120575119909V119899119894+12 = 12 (120575119909V
119899+1119894+12 + 120575119909V
119899minus1119894+12)
1205752119909V
119899119894+12 = 1ℎ (120575119909V
119899119894+12 minus 120575119909V
119899119894minus12)
(16)
from which
(120575119909V119898119894+12) V119899119894+12 + V119898119894+12120575119909V
119899119894+12 = 120575119909 (V119898119894+12V
119899119894+12) (17)
The discrete system is defined by
nabla119905119888119899119894 minus 1ℎ [(119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12)minus (119888119899119894minus12120575119909120579119899
119894minus12 + 120579119899119894minus12120575119909119888119899119894minus12)] = minusΓ (119888119899119894 120579119899
119894 ) 1 le 119894 le 119872 minus 1
(18)
nabla1199051198881198990 minus 2ℎ [(1198881198991212057511990912057911989912 + 120579119899
1212057511990911988811989912) minus 1205721 (1198881198990 minus 1205831)]= minusΓ (1198881198990 120579119899
0) (19)
nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872)minus (119888119899119872minus12120575119909120579119899
119872minus12 + 120579119899119872minus12120575119909119888119899119872minus12)]
= minusΓ (119888119899119872 120579119899119872)
(20)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581ℎ (120575119909120579119899119894+12 minus 120575119909120579119899
119894minus12) = (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) 1 le 119894 le 119872 minus 1
(21)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [120575119909120579119899
12
minus 1205731 (1205791198990 minus ]1)] = (120582 + 120579119899
0) Γ (1198881198990 1205791198990)
(22)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 120575119909120579119899
119872minus12] = (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872) 1 le 119899 le 119873 minus 1
(23)
and the discrete initial conditions
1198880119894 = 1198880 (119909119894) 1205790119894 = 1205790 (119909119894) (24)
1198881119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) 1205791119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) 0 le 119894 le 119872 (25)
where
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12 (26)
The computational procedure of the uncoupled leap-frogscheme at each time step is listed below
Step 1 The vapor concentration 119888119899+1119895 can be calculated by
solving the tridiagonal linear systems defined in (18)-(20)
Step 2 With the updated vapor concentration 119888119899+1119895 we can
get 1198881198990 and 119906119899119894+12 correspondingly
Step 3 Finally the temperature 120579119899+1119895 can be obtained by
solving the tridiagonal linear system (21)-(23)
3 The Leap-Frog Scheme andthe Optimal Error Estimate
In this section we will show the existence and uniquenessof the solution to the system (18)-(26) with optimal errorestimates in the discrete 1198712 norm Let V = V119899119895 119872119895=0 and 119911 =119911119895119872119895=0 be two mesh functions on Ωℎ We define the innerproduct and norms by
(V 119911) = ℎ(12V01199110 +119872minus1sum119894=1
V119894119911119894 + 12V119872119911119872) V = radicℎ[12 (V0)2 +
119872minus1sum119894=1
(V119894)2 + 12 (V119872)2]Vinfin = max
0le119894le119872
1003816100381610038161003816V1198941003816100381610038161003816 1003817100381710038171003817120575119909V
1003817100381710038171003817 = radicℎ119872minus1sum119894=0
(V119894+1 minus V119894ℎ )2100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 = radicℎ119872minus1sum
119894=1
(1205752119909V119894)2
(27)
Let (119862 Θ) be the solution of the system (6)-(12) and 119862119899119894 =119888(119909119894 119905119899) Θ119899
119894 = 120579(119909119894 119905119899) The error functions are defined by
119888119899119894 = 119862119899119894 minus 119888119899119894
120579119899119894 = Θ119899
119894 minus 120579119899119894 0 le 119894 le 119872 0 le 119899 le 119873
(28)
We state our main result in the theorem below
Theorem 1 Suppose that the solution (119888 V) of the system (6)-(12) is in 11986243([0 119871] times [0T]) satisfying (13) en there existpositive constants ℎ0 and 1198640 independent of ℎ and 120591 such that
4 Journal of Mathematics
when 120591 le 119864119888ℎ le ℎ0 the finite difference scheme (18)-(26) isuniquely solvable and
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 119899sum119898=1
(10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172)le 1198640 (1205912 + ℎ2)2 1 le 119899 le 119873 minus 1
(29)
To prove the theorem we make a stronger assumptionthat there exists 1198640 gt 0 independent of 119899 ℎ 120591 such that theinequality
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (30)
holds for 119899 le 119896minus1We prove the assumption and the theoremby inductionmethod By the initial condition (26) this is truefor 119899 = 0 In the next subsection we will show that this is alsotrue for 119899 = 1 In this part we let 119864119888 be a generic positiveconstant which is associated with the physical parameters 120590120582 120581 119888min 119888max 120579min 120579max the parameters involved in initialand boundary conditions and the solution of the system (6)-(12) 119864119888 is independent of time step 119899 mesh size ℎ 120591 andconstant 1198640
31 e Leap-Frog Scheme and Preliminaries For conve-nience of calculations we further introduce some notationsLet 119906 = (119888120579)119909 119908 = 120579119909 thus the sweat transport system (6)-(7)can be reduced to
119888119905 minus 119906119909 = minusΓ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (31)
119906 = (119888120579119909 + 120579119888119909) 0 le 119909 le 119871 0 lt 119905 le T (32)
(119888 + 120590) 120579119905 minus 119906120579119909 minus 120581119908119909 = (120582 + 120579) Γ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (33)
119908 = 120579119909 0 le 119909 le 119871 0 lt 119905 le T (34)
with the initial and boundary conditions
119906 = 1205721 (119888 minus 1205831) 119909 = 0 0 lt 119905 le T119906 = 1205722 (1205832 minus 119888) 119909 = 119871 0 lt 119905 le T119908 = 1205731 (120579 minus ]1) 119909 = 0 0 lt 119905 le T119908 = 1205732 (]2 minus 120579) 119909 = 119871 0 lt 119905 le T
119888 (119909 0) = 1198880 (119909) 120579 (119909 0) = 1205790 (119909) 0 le 119909 le 119871
(35)
The discrete leap-frog system (18)-(23) is modified as
nabla119905119888119899119894 minus 120575lowast119909119906119899
119894 = minusΓ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
nabla1199051198881198990 minus 2ℎ [11990611989912 minus 1205721 (1198881198990 minus 1205831)] = minusΓ (1198881198990 120579119899
0) nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872) minus 119906119899
119872minus12] = minusΓ (119888119899119872 120579119899119872)
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12
0 le 119894 le 119872 minus 1(119888119899119894 + 120590)nabla119905120579119899
119894 minus 12 (119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12)minus 120581120575lowast
119909119908119899119894 = (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912
minus 2120581ℎ [11990811989912 minus 1205731 (120579119899
0 minus ]1)] = (120582 + 1205791198990) Γ (1198881198990 120579119899
0) (119888119899119872 + 120590)nabla119905120579119899
119872 minus 119906119899119872minus12120575119909120579119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 119908119899
119872minus12]= (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)
119908119899119894+12 = 120575119909120579119899
119894+12 0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(36)
Let 119880119899119894 = 119906(119909119894 119905119899) = (119888120579)119909(119909119894 119905119899) and 119882119899
119894 = 119908(119909119894 119905119899) =120579119909(119909119894 119905119899) We denote by 119906119899119894 and 119908119899
119894 the corresponding finitedifference solution and
119899119894+12 = 119880119899
119894+12 minus 119906119899119894+12
119908119899119894+12 = 119882119899
119894+12 minus 119908119899119894+12
0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1(37)
We get
nabla119905119862119899119894 minus 120575lowast
119909119880119899119894 = minusΓ (119862119899
119894 Θ119899119894 ) + 119877119899
119888119894 1 le 119894 le 119872 minus 1nabla119905119862119899
0 minus 2ℎ [11988011989912 minus 1205721 (119862119899
0 minus 1205831)] = minusΓ (1198621198990 Θ119899
0) + 1198771198991198880
nabla119905119862119899119872 minus 2ℎ [1205722 (1205832 minus 119862119899
119872) minus 119880119899119872minus12] = minusΓ (119862119899
119872 Θ119899119872)
+ 119877119899119888119872
119880119899119894+12 = 119862119899
119894+12120575119909Θ119899119894+12 + Θ119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+120 le 119894 le 119872 minus 1
(119862119899119894 + 120590)nabla119905Θ119899
119894 minus 12 (119880119899119894minus12120575119909Θ119899
119894minus12 + 119880119899119894+12120575119909Θ119899
119894+12)minus 120581120575lowast
119909119882119899119894 = (120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) + 1198771198991205791198941 le 119894 le 119872 minus 1
Journal of Mathematics 5
(1198621198990 + 120590)nabla119905Θ119899
0 minus 11988011989912120575119909Θ119899
12
minus 2120581ℎ [11988211989912 minus 1205731 (Θ119899
0 minus ]1)]= (120582 + Θ119899
0) Γ (1198621198990 Θ119899
0) + 1198771198991205790
(119862119899119872 + 120590)nabla119905Θ119899
119872 minus 119880119899119872minus12120575119909Θ119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus Θ119899119872) minus119882119899
119872minus12]= (120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) + 119877119899120579119872
119882119899119894+12 = 120575119909Θ119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(38)
and the initial conditions
1198620119894 = 1198880 (119909119894)
Θ0119894 = 1205790 (119909119894)
1198621119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) + 1205911198771
119888119894Θ1
119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) + 1205911198771120579119894
(39)
where 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 le 1198641 (120591 + ℎ2) 0 le 119894 le 11987210038161003816100381610038161003816119877119899119888119894
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119894
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 1 le 119894 le 119872 minus 110038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119908119894+12
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 0 le 119894 le 119872 minus 1100381610038161003816100381610038161198771198991198880
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119888119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 100381610038161003816100381610038161198771198991205790
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ) 1 le 119899 le 119873 minus 1
(40)
Subtracting the system (36) from the system (38) we get theerror equations
nabla119905119888119899119894 minus 120575lowast119909 119899
119894 = minus [Γ (119862119899119894 Θ119899
119894 ) minus Γ (119888119899119894 120579119899119894 )] + 119877119899
119888119894
fl 119877119899
119888119894 1 le 119894 le 119872 minus 1 (41)
nabla1199051198881198990 minus 2ℎ [11989912 minus 12057211198881198990 ] = minus [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)]+ 119877119899
1198880 fl 119877119899
1198880(42)
nabla119905119888119899119872 minus 2ℎ [minus1205722119888119899119872 minus 119899119872minus12]
= minus [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] + 119877119899
119888119872 fl 119877119899
119888119872(43)
119899119894+12 = 119888119899119894+12120575119909Θ119899
119894+12 + 120579119899119894+12120575119909119862119899
119894+12
+ 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12 + 119877119899119906119894+120 le 119894 le 119872 minus 1
(44)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581120575lowast119909119908119899
119894 = minus119888119899119894 nabla119905Θ119899119894
+ 12 (119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)+ [(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )]+ 119877119899
120579119894 fl 119877119899
120579119894 1 le 119894 le 119872 minus 1
(45)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [119908119899
12 minus 12057311205791198990]
= minus1198881198990nabla119905Θ1198990 + 119899
12120575119909Θ11989912
+ [(120582 + Θ1198990) Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)]
+ 1198771198991205790 fl 119877119899
1205790(46)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [minus1205732120579119899119872 minus 119908119899
119872minus12] = minus119888119899119872nabla119905Θ119899119872
+ 119899119872minus12120575119909Θ119899
119872minus12
+ [(120582 + Θ119899119872) Γ (119862119899
119872 Θ119899119872) minus (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)]
+ 119877119899120579119872 fl 119877119899
120579119872
(47)
119908119899119894+12 = 120575119909120579119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1 (48)
and
1198880119894 = 01205790119894 = 01198881119894 = 1205911198771
1198881198941205791119894 = 1205911198771
120579119894(49)
and by (40) we can directly derive the inequality
100381710038171003817100381710038171198881100381710038171003817100381710038172 + 100381710038171003817100381710038171205791100381710038171003817100381710038172 + 100381710038171003817100381710038171205751199091205791100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (50)
To prove our main theorem the following formula will beoften used
[V121199110 + ℎ119872minus1sum119894=1
120575119909V119894119911119894 minus V119872minus12119911119872]= 119872minus1sum
119894=0
V119894+12 (119911119896119894 minus 119911119896
119894+1) (51)
In the following lemma we present discrete Sobolev interpo-lation formulas and the proof can be found in [24]
6 Journal of Mathematics
Lemma 2 Let V and 119911 be two mesh functions en for anypositive constant 120598
V2infin le 120598 1003817100381710038171003817120575119909V10038171003817100381710038172 + (1120598 + 1119871) V2 (52)
1003817100381710038171003817120575119909V1003817100381710038171003817infin le 120598 100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 + 119864119888
1003817100381710038171003817120575119909V10038171003817100381710038172 (53)
Lemma 3
10038171003817100381710038171198881198991003817100381710038171003817infin 1003817100381710038171003817100381712057911989910038171003817100381710038171003817infin le 3119864120 (12059174 + ℎ32) 1 le 119899 le 119896 (54)1003817100381710038171003817100381711988811989911990910038171003817100381710038171003817 10038171003817100381710038171003817120579119899
119909
10038171003817100381710038171003817 le 119864120 (12059132 + ℎ) 1 le 119899 le 119896 minus 1 (55)
Proof From (30) for 0 le 119899 le 119896 minus 1 we have10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (56)
When 120591 le ℎ with the inverse inequality we have
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infin le 2ℎminus1 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 le 2ℎminus11198640 (1205912 + ℎ2)2le 81198640ℎ3 (57)
When ℎ le 120591 by taking 120598 = 12059112 in Lemma 2
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infinle 12059112 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 + (120591minus12 + 119871minus1) 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172le (120591minus12 + 12059112119871minus1) 1198640 (1205912 + ℎ2)2 le 8119864012059172
1 le 119894 le 119872 minus 1 0 le 119899 le 119896 minus 1(58)
The first part of (54) is obtained and the second part and theinequality (55) can be proved similarly
In addition by Lemma 3 there exist constants 1198643 gt 0 and1199040 gt 0 such that when ℎ 120591 le 1199040119888min2 le 119888119899119894 le 2119888max120579min2 le 120579119899
119894 le 2120579max0 le 119894 le 119872 minus 1 1 le 119899 le 119896
(59)
and 1003816100381610038161003816nabla119905Θ1198991198941003816100381610038161003816 1003816100381610038161003816nabla119905119862119899
1198941003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816120575119909119862119899
119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119862119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 le 11986430 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816119888119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120579119899119894+12
10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 119896
10038161003816100381610038161003816Γ (119888119899119894+12 120579119899119894+12)10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 1198961003816100381610038161003816(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899
119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 1198961003816100381610038161003816Γ (119862119899
119894 Θ119899119894 ) minus Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816 le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 119896(60)
32 e Existence and Uniqueness Since the coefficientmatrix in the system (18)-(20) is strictly diagonally dominantthus the system (18)-(20) has a unique solution 119888119896+1
119894 Here wewill discuss the boundedness of 119888119896+1
119894 Multiplying (41)-(43) by ℎ119888119899119894 ℎ1198881198990 2 and ℎ1198881198991198722 respec-
tively we get
(nabla119905119888119899 119888119899) + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162 = [119899121198881198990
+ ℎ119872minus1sum119894=1
120575lowast119909 119899
119894 119888119899119894 + 119899119872minus12119888119899119872] + ℎ119872minus1sum
119894=1
119877119899119888119894119888119899119894 + ℎ2
sdot 11987711989911988801198881198990 + ℎ2119877119899
119888119872119888119899119872minus ℎ12 [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)] 1198881198990+ 119872minus1sum
119894=1
[Γ (119862119899119894 Θn
119894 ) minus Γ (119888119899119894 120579119899119894 )] 119888119899119894
+ 12 [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] 119888119899119872 fl 1198691 + 1198692
+ 1198693 1 le 119899 le 119896
(61)
With (44) (51) (59) (60) and (40) we see that
minus 1198691 = minus119899121198881198990 minus ℎ119872minus1sum
119894=1
120575119909119899119894 119888119899119894 + 119899
Mminus12119888119899119872= ℎ119872minus1sum
119894=0
119899119894+12120575119909119888119899119894+12 = ℎ119872minus1sum
119894=0
[119888119899119894+12120575119909Θ119899119894+12
+ 120579119899119894+12120575119909119862119899
119894+12 + 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119877119899119906119894+12] 120575119909119888119899119894+12 ge 120579min2 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 2119888max
1003817100381710038171003817100381712057511990912057911989910038171003817100381710038171003817sdot 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817 minus 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817minus ℎ119872minus1sum
119894=0
10038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119888119899119894+12
10038161003816100381610038161003816 ge 120579min4 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172minus 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(62)
Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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Journal of Mathematics 3
120575119909V119899119894+12 = 1ℎ (V119899119894+1 minus V119899119894 ) 120575lowast119909V
119899119894 = 1ℎ (V119899119894+12 minus V119899119894minus12)
120575119909V119899119894+12 = 12 (120575119909V
119899+1119894+12 + 120575119909V
119899minus1119894+12)
1205752119909V
119899119894+12 = 1ℎ (120575119909V
119899119894+12 minus 120575119909V
119899119894minus12)
(16)
from which
(120575119909V119898119894+12) V119899119894+12 + V119898119894+12120575119909V
119899119894+12 = 120575119909 (V119898119894+12V
119899119894+12) (17)
The discrete system is defined by
nabla119905119888119899119894 minus 1ℎ [(119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12)minus (119888119899119894minus12120575119909120579119899
119894minus12 + 120579119899119894minus12120575119909119888119899119894minus12)] = minusΓ (119888119899119894 120579119899
119894 ) 1 le 119894 le 119872 minus 1
(18)
nabla1199051198881198990 minus 2ℎ [(1198881198991212057511990912057911989912 + 120579119899
1212057511990911988811989912) minus 1205721 (1198881198990 minus 1205831)]= minusΓ (1198881198990 120579119899
0) (19)
nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872)minus (119888119899119872minus12120575119909120579119899
119872minus12 + 120579119899119872minus12120575119909119888119899119872minus12)]
= minusΓ (119888119899119872 120579119899119872)
(20)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581ℎ (120575119909120579119899119894+12 minus 120575119909120579119899
119894minus12) = (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) 1 le 119894 le 119872 minus 1
(21)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [120575119909120579119899
12
minus 1205731 (1205791198990 minus ]1)] = (120582 + 120579119899
0) Γ (1198881198990 1205791198990)
(22)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 120575119909120579119899
119872minus12] = (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872) 1 le 119899 le 119873 minus 1
(23)
and the discrete initial conditions
1198880119894 = 1198880 (119909119894) 1205790119894 = 1205790 (119909119894) (24)
1198881119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) 1205791119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) 0 le 119894 le 119872 (25)
where
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12 (26)
The computational procedure of the uncoupled leap-frogscheme at each time step is listed below
Step 1 The vapor concentration 119888119899+1119895 can be calculated by
solving the tridiagonal linear systems defined in (18)-(20)
Step 2 With the updated vapor concentration 119888119899+1119895 we can
get 1198881198990 and 119906119899119894+12 correspondingly
Step 3 Finally the temperature 120579119899+1119895 can be obtained by
solving the tridiagonal linear system (21)-(23)
3 The Leap-Frog Scheme andthe Optimal Error Estimate
In this section we will show the existence and uniquenessof the solution to the system (18)-(26) with optimal errorestimates in the discrete 1198712 norm Let V = V119899119895 119872119895=0 and 119911 =119911119895119872119895=0 be two mesh functions on Ωℎ We define the innerproduct and norms by
(V 119911) = ℎ(12V01199110 +119872minus1sum119894=1
V119894119911119894 + 12V119872119911119872) V = radicℎ[12 (V0)2 +
119872minus1sum119894=1
(V119894)2 + 12 (V119872)2]Vinfin = max
0le119894le119872
1003816100381610038161003816V1198941003816100381610038161003816 1003817100381710038171003817120575119909V
1003817100381710038171003817 = radicℎ119872minus1sum119894=0
(V119894+1 minus V119894ℎ )2100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 = radicℎ119872minus1sum
119894=1
(1205752119909V119894)2
(27)
Let (119862 Θ) be the solution of the system (6)-(12) and 119862119899119894 =119888(119909119894 119905119899) Θ119899
119894 = 120579(119909119894 119905119899) The error functions are defined by
119888119899119894 = 119862119899119894 minus 119888119899119894
120579119899119894 = Θ119899
119894 minus 120579119899119894 0 le 119894 le 119872 0 le 119899 le 119873
(28)
We state our main result in the theorem below
Theorem 1 Suppose that the solution (119888 V) of the system (6)-(12) is in 11986243([0 119871] times [0T]) satisfying (13) en there existpositive constants ℎ0 and 1198640 independent of ℎ and 120591 such that
4 Journal of Mathematics
when 120591 le 119864119888ℎ le ℎ0 the finite difference scheme (18)-(26) isuniquely solvable and
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 119899sum119898=1
(10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172)le 1198640 (1205912 + ℎ2)2 1 le 119899 le 119873 minus 1
(29)
To prove the theorem we make a stronger assumptionthat there exists 1198640 gt 0 independent of 119899 ℎ 120591 such that theinequality
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (30)
holds for 119899 le 119896minus1We prove the assumption and the theoremby inductionmethod By the initial condition (26) this is truefor 119899 = 0 In the next subsection we will show that this is alsotrue for 119899 = 1 In this part we let 119864119888 be a generic positiveconstant which is associated with the physical parameters 120590120582 120581 119888min 119888max 120579min 120579max the parameters involved in initialand boundary conditions and the solution of the system (6)-(12) 119864119888 is independent of time step 119899 mesh size ℎ 120591 andconstant 1198640
31 e Leap-Frog Scheme and Preliminaries For conve-nience of calculations we further introduce some notationsLet 119906 = (119888120579)119909 119908 = 120579119909 thus the sweat transport system (6)-(7)can be reduced to
119888119905 minus 119906119909 = minusΓ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (31)
119906 = (119888120579119909 + 120579119888119909) 0 le 119909 le 119871 0 lt 119905 le T (32)
(119888 + 120590) 120579119905 minus 119906120579119909 minus 120581119908119909 = (120582 + 120579) Γ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (33)
119908 = 120579119909 0 le 119909 le 119871 0 lt 119905 le T (34)
with the initial and boundary conditions
119906 = 1205721 (119888 minus 1205831) 119909 = 0 0 lt 119905 le T119906 = 1205722 (1205832 minus 119888) 119909 = 119871 0 lt 119905 le T119908 = 1205731 (120579 minus ]1) 119909 = 0 0 lt 119905 le T119908 = 1205732 (]2 minus 120579) 119909 = 119871 0 lt 119905 le T
119888 (119909 0) = 1198880 (119909) 120579 (119909 0) = 1205790 (119909) 0 le 119909 le 119871
(35)
The discrete leap-frog system (18)-(23) is modified as
nabla119905119888119899119894 minus 120575lowast119909119906119899
119894 = minusΓ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
nabla1199051198881198990 minus 2ℎ [11990611989912 minus 1205721 (1198881198990 minus 1205831)] = minusΓ (1198881198990 120579119899
0) nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872) minus 119906119899
119872minus12] = minusΓ (119888119899119872 120579119899119872)
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12
0 le 119894 le 119872 minus 1(119888119899119894 + 120590)nabla119905120579119899
119894 minus 12 (119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12)minus 120581120575lowast
119909119908119899119894 = (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912
minus 2120581ℎ [11990811989912 minus 1205731 (120579119899
0 minus ]1)] = (120582 + 1205791198990) Γ (1198881198990 120579119899
0) (119888119899119872 + 120590)nabla119905120579119899
119872 minus 119906119899119872minus12120575119909120579119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 119908119899
119872minus12]= (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)
119908119899119894+12 = 120575119909120579119899
119894+12 0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(36)
Let 119880119899119894 = 119906(119909119894 119905119899) = (119888120579)119909(119909119894 119905119899) and 119882119899
119894 = 119908(119909119894 119905119899) =120579119909(119909119894 119905119899) We denote by 119906119899119894 and 119908119899
119894 the corresponding finitedifference solution and
119899119894+12 = 119880119899
119894+12 minus 119906119899119894+12
119908119899119894+12 = 119882119899
119894+12 minus 119908119899119894+12
0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1(37)
We get
nabla119905119862119899119894 minus 120575lowast
119909119880119899119894 = minusΓ (119862119899
119894 Θ119899119894 ) + 119877119899
119888119894 1 le 119894 le 119872 minus 1nabla119905119862119899
0 minus 2ℎ [11988011989912 minus 1205721 (119862119899
0 minus 1205831)] = minusΓ (1198621198990 Θ119899
0) + 1198771198991198880
nabla119905119862119899119872 minus 2ℎ [1205722 (1205832 minus 119862119899
119872) minus 119880119899119872minus12] = minusΓ (119862119899
119872 Θ119899119872)
+ 119877119899119888119872
119880119899119894+12 = 119862119899
119894+12120575119909Θ119899119894+12 + Θ119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+120 le 119894 le 119872 minus 1
(119862119899119894 + 120590)nabla119905Θ119899
119894 minus 12 (119880119899119894minus12120575119909Θ119899
119894minus12 + 119880119899119894+12120575119909Θ119899
119894+12)minus 120581120575lowast
119909119882119899119894 = (120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) + 1198771198991205791198941 le 119894 le 119872 minus 1
Journal of Mathematics 5
(1198621198990 + 120590)nabla119905Θ119899
0 minus 11988011989912120575119909Θ119899
12
minus 2120581ℎ [11988211989912 minus 1205731 (Θ119899
0 minus ]1)]= (120582 + Θ119899
0) Γ (1198621198990 Θ119899
0) + 1198771198991205790
(119862119899119872 + 120590)nabla119905Θ119899
119872 minus 119880119899119872minus12120575119909Θ119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus Θ119899119872) minus119882119899
119872minus12]= (120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) + 119877119899120579119872
119882119899119894+12 = 120575119909Θ119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(38)
and the initial conditions
1198620119894 = 1198880 (119909119894)
Θ0119894 = 1205790 (119909119894)
1198621119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) + 1205911198771
119888119894Θ1
119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) + 1205911198771120579119894
(39)
where 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 le 1198641 (120591 + ℎ2) 0 le 119894 le 11987210038161003816100381610038161003816119877119899119888119894
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119894
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 1 le 119894 le 119872 minus 110038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119908119894+12
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 0 le 119894 le 119872 minus 1100381610038161003816100381610038161198771198991198880
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119888119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 100381610038161003816100381610038161198771198991205790
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ) 1 le 119899 le 119873 minus 1
(40)
Subtracting the system (36) from the system (38) we get theerror equations
nabla119905119888119899119894 minus 120575lowast119909 119899
119894 = minus [Γ (119862119899119894 Θ119899
119894 ) minus Γ (119888119899119894 120579119899119894 )] + 119877119899
119888119894
fl 119877119899
119888119894 1 le 119894 le 119872 minus 1 (41)
nabla1199051198881198990 minus 2ℎ [11989912 minus 12057211198881198990 ] = minus [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)]+ 119877119899
1198880 fl 119877119899
1198880(42)
nabla119905119888119899119872 minus 2ℎ [minus1205722119888119899119872 minus 119899119872minus12]
= minus [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] + 119877119899
119888119872 fl 119877119899
119888119872(43)
119899119894+12 = 119888119899119894+12120575119909Θ119899
119894+12 + 120579119899119894+12120575119909119862119899
119894+12
+ 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12 + 119877119899119906119894+120 le 119894 le 119872 minus 1
(44)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581120575lowast119909119908119899
119894 = minus119888119899119894 nabla119905Θ119899119894
+ 12 (119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)+ [(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )]+ 119877119899
120579119894 fl 119877119899
120579119894 1 le 119894 le 119872 minus 1
(45)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [119908119899
12 minus 12057311205791198990]
= minus1198881198990nabla119905Θ1198990 + 119899
12120575119909Θ11989912
+ [(120582 + Θ1198990) Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)]
+ 1198771198991205790 fl 119877119899
1205790(46)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [minus1205732120579119899119872 minus 119908119899
119872minus12] = minus119888119899119872nabla119905Θ119899119872
+ 119899119872minus12120575119909Θ119899
119872minus12
+ [(120582 + Θ119899119872) Γ (119862119899
119872 Θ119899119872) minus (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)]
+ 119877119899120579119872 fl 119877119899
120579119872
(47)
119908119899119894+12 = 120575119909120579119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1 (48)
and
1198880119894 = 01205790119894 = 01198881119894 = 1205911198771
1198881198941205791119894 = 1205911198771
120579119894(49)
and by (40) we can directly derive the inequality
100381710038171003817100381710038171198881100381710038171003817100381710038172 + 100381710038171003817100381710038171205791100381710038171003817100381710038172 + 100381710038171003817100381710038171205751199091205791100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (50)
To prove our main theorem the following formula will beoften used
[V121199110 + ℎ119872minus1sum119894=1
120575119909V119894119911119894 minus V119872minus12119911119872]= 119872minus1sum
119894=0
V119894+12 (119911119896119894 minus 119911119896
119894+1) (51)
In the following lemma we present discrete Sobolev interpo-lation formulas and the proof can be found in [24]
6 Journal of Mathematics
Lemma 2 Let V and 119911 be two mesh functions en for anypositive constant 120598
V2infin le 120598 1003817100381710038171003817120575119909V10038171003817100381710038172 + (1120598 + 1119871) V2 (52)
1003817100381710038171003817120575119909V1003817100381710038171003817infin le 120598 100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 + 119864119888
1003817100381710038171003817120575119909V10038171003817100381710038172 (53)
Lemma 3
10038171003817100381710038171198881198991003817100381710038171003817infin 1003817100381710038171003817100381712057911989910038171003817100381710038171003817infin le 3119864120 (12059174 + ℎ32) 1 le 119899 le 119896 (54)1003817100381710038171003817100381711988811989911990910038171003817100381710038171003817 10038171003817100381710038171003817120579119899
119909
10038171003817100381710038171003817 le 119864120 (12059132 + ℎ) 1 le 119899 le 119896 minus 1 (55)
Proof From (30) for 0 le 119899 le 119896 minus 1 we have10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (56)
When 120591 le ℎ with the inverse inequality we have
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infin le 2ℎminus1 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 le 2ℎminus11198640 (1205912 + ℎ2)2le 81198640ℎ3 (57)
When ℎ le 120591 by taking 120598 = 12059112 in Lemma 2
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infinle 12059112 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 + (120591minus12 + 119871minus1) 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172le (120591minus12 + 12059112119871minus1) 1198640 (1205912 + ℎ2)2 le 8119864012059172
1 le 119894 le 119872 minus 1 0 le 119899 le 119896 minus 1(58)
The first part of (54) is obtained and the second part and theinequality (55) can be proved similarly
In addition by Lemma 3 there exist constants 1198643 gt 0 and1199040 gt 0 such that when ℎ 120591 le 1199040119888min2 le 119888119899119894 le 2119888max120579min2 le 120579119899
119894 le 2120579max0 le 119894 le 119872 minus 1 1 le 119899 le 119896
(59)
and 1003816100381610038161003816nabla119905Θ1198991198941003816100381610038161003816 1003816100381610038161003816nabla119905119862119899
1198941003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816120575119909119862119899
119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119862119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 le 11986430 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816119888119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120579119899119894+12
10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 119896
10038161003816100381610038161003816Γ (119888119899119894+12 120579119899119894+12)10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 1198961003816100381610038161003816(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899
119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 1198961003816100381610038161003816Γ (119862119899
119894 Θ119899119894 ) minus Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816 le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 119896(60)
32 e Existence and Uniqueness Since the coefficientmatrix in the system (18)-(20) is strictly diagonally dominantthus the system (18)-(20) has a unique solution 119888119896+1
119894 Here wewill discuss the boundedness of 119888119896+1
119894 Multiplying (41)-(43) by ℎ119888119899119894 ℎ1198881198990 2 and ℎ1198881198991198722 respec-
tively we get
(nabla119905119888119899 119888119899) + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162 = [119899121198881198990
+ ℎ119872minus1sum119894=1
120575lowast119909 119899
119894 119888119899119894 + 119899119872minus12119888119899119872] + ℎ119872minus1sum
119894=1
119877119899119888119894119888119899119894 + ℎ2
sdot 11987711989911988801198881198990 + ℎ2119877119899
119888119872119888119899119872minus ℎ12 [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)] 1198881198990+ 119872minus1sum
119894=1
[Γ (119862119899119894 Θn
119894 ) minus Γ (119888119899119894 120579119899119894 )] 119888119899119894
+ 12 [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] 119888119899119872 fl 1198691 + 1198692
+ 1198693 1 le 119899 le 119896
(61)
With (44) (51) (59) (60) and (40) we see that
minus 1198691 = minus119899121198881198990 minus ℎ119872minus1sum
119894=1
120575119909119899119894 119888119899119894 + 119899
Mminus12119888119899119872= ℎ119872minus1sum
119894=0
119899119894+12120575119909119888119899119894+12 = ℎ119872minus1sum
119894=0
[119888119899119894+12120575119909Θ119899119894+12
+ 120579119899119894+12120575119909119862119899
119894+12 + 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119877119899119906119894+12] 120575119909119888119899119894+12 ge 120579min2 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 2119888max
1003817100381710038171003817100381712057511990912057911989910038171003817100381710038171003817sdot 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817 minus 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817minus ℎ119872minus1sum
119894=0
10038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119888119899119894+12
10038161003816100381610038161003816 ge 120579min4 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172minus 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(62)
Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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4 Journal of Mathematics
when 120591 le 119864119888ℎ le ℎ0 the finite difference scheme (18)-(26) isuniquely solvable and
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 119899sum119898=1
(10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172)le 1198640 (1205912 + ℎ2)2 1 le 119899 le 119873 minus 1
(29)
To prove the theorem we make a stronger assumptionthat there exists 1198640 gt 0 independent of 119899 ℎ 120591 such that theinequality
10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (30)
holds for 119899 le 119896minus1We prove the assumption and the theoremby inductionmethod By the initial condition (26) this is truefor 119899 = 0 In the next subsection we will show that this is alsotrue for 119899 = 1 In this part we let 119864119888 be a generic positiveconstant which is associated with the physical parameters 120590120582 120581 119888min 119888max 120579min 120579max the parameters involved in initialand boundary conditions and the solution of the system (6)-(12) 119864119888 is independent of time step 119899 mesh size ℎ 120591 andconstant 1198640
31 e Leap-Frog Scheme and Preliminaries For conve-nience of calculations we further introduce some notationsLet 119906 = (119888120579)119909 119908 = 120579119909 thus the sweat transport system (6)-(7)can be reduced to
119888119905 minus 119906119909 = minusΓ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (31)
119906 = (119888120579119909 + 120579119888119909) 0 le 119909 le 119871 0 lt 119905 le T (32)
(119888 + 120590) 120579119905 minus 119906120579119909 minus 120581119908119909 = (120582 + 120579) Γ (119888 120579) 0 le 119909 le 119871 0 lt 119905 le T (33)
119908 = 120579119909 0 le 119909 le 119871 0 lt 119905 le T (34)
with the initial and boundary conditions
119906 = 1205721 (119888 minus 1205831) 119909 = 0 0 lt 119905 le T119906 = 1205722 (1205832 minus 119888) 119909 = 119871 0 lt 119905 le T119908 = 1205731 (120579 minus ]1) 119909 = 0 0 lt 119905 le T119908 = 1205732 (]2 minus 120579) 119909 = 119871 0 lt 119905 le T
119888 (119909 0) = 1198880 (119909) 120579 (119909 0) = 1205790 (119909) 0 le 119909 le 119871
(35)
The discrete leap-frog system (18)-(23) is modified as
nabla119905119888119899119894 minus 120575lowast119909119906119899
119894 = minusΓ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
nabla1199051198881198990 minus 2ℎ [11990611989912 minus 1205721 (1198881198990 minus 1205831)] = minusΓ (1198881198990 120579119899
0) nabla119905119888119899119872 minus 2ℎ [1205722 (1205832 minus 119888119899119872) minus 119906119899
119872minus12] = minusΓ (119888119899119872 120579119899119872)
119906119899119894+12 = 119888119899119894+12120575119909120579119899
119894+12 + 120579119899119894+12120575119909119888119899119894+12
0 le 119894 le 119872 minus 1(119888119899119894 + 120590)nabla119905120579119899
119894 minus 12 (119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12)minus 120581120575lowast
119909119908119899119894 = (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 ) 1 le 119894 le 119872 minus 1
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912
minus 2120581ℎ [11990811989912 minus 1205731 (120579119899
0 minus ]1)] = (120582 + 1205791198990) Γ (1198881198990 120579119899
0) (119888119899119872 + 120590)nabla119905120579119899
119872 minus 119906119899119872minus12120575119909120579119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus 120579119899119872) minus 119908119899
119872minus12]= (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)
119908119899119894+12 = 120575119909120579119899
119894+12 0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(36)
Let 119880119899119894 = 119906(119909119894 119905119899) = (119888120579)119909(119909119894 119905119899) and 119882119899
119894 = 119908(119909119894 119905119899) =120579119909(119909119894 119905119899) We denote by 119906119899119894 and 119908119899
119894 the corresponding finitedifference solution and
119899119894+12 = 119880119899
119894+12 minus 119906119899119894+12
119908119899119894+12 = 119882119899
119894+12 minus 119908119899119894+12
0 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1(37)
We get
nabla119905119862119899119894 minus 120575lowast
119909119880119899119894 = minusΓ (119862119899
119894 Θ119899119894 ) + 119877119899
119888119894 1 le 119894 le 119872 minus 1nabla119905119862119899
0 minus 2ℎ [11988011989912 minus 1205721 (119862119899
0 minus 1205831)] = minusΓ (1198621198990 Θ119899
0) + 1198771198991198880
nabla119905119862119899119872 minus 2ℎ [1205722 (1205832 minus 119862119899
119872) minus 119880119899119872minus12] = minusΓ (119862119899
119872 Θ119899119872)
+ 119877119899119888119872
119880119899119894+12 = 119862119899
119894+12120575119909Θ119899119894+12 + Θ119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+120 le 119894 le 119872 minus 1
(119862119899119894 + 120590)nabla119905Θ119899
119894 minus 12 (119880119899119894minus12120575119909Θ119899
119894minus12 + 119880119899119894+12120575119909Θ119899
119894+12)minus 120581120575lowast
119909119882119899119894 = (120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) + 1198771198991205791198941 le 119894 le 119872 minus 1
Journal of Mathematics 5
(1198621198990 + 120590)nabla119905Θ119899
0 minus 11988011989912120575119909Θ119899
12
minus 2120581ℎ [11988211989912 minus 1205731 (Θ119899
0 minus ]1)]= (120582 + Θ119899
0) Γ (1198621198990 Θ119899
0) + 1198771198991205790
(119862119899119872 + 120590)nabla119905Θ119899
119872 minus 119880119899119872minus12120575119909Θ119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus Θ119899119872) minus119882119899
119872minus12]= (120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) + 119877119899120579119872
119882119899119894+12 = 120575119909Θ119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(38)
and the initial conditions
1198620119894 = 1198880 (119909119894)
Θ0119894 = 1205790 (119909119894)
1198621119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) + 1205911198771
119888119894Θ1
119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) + 1205911198771120579119894
(39)
where 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 le 1198641 (120591 + ℎ2) 0 le 119894 le 11987210038161003816100381610038161003816119877119899119888119894
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119894
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 1 le 119894 le 119872 minus 110038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119908119894+12
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 0 le 119894 le 119872 minus 1100381610038161003816100381610038161198771198991198880
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119888119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 100381610038161003816100381610038161198771198991205790
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ) 1 le 119899 le 119873 minus 1
(40)
Subtracting the system (36) from the system (38) we get theerror equations
nabla119905119888119899119894 minus 120575lowast119909 119899
119894 = minus [Γ (119862119899119894 Θ119899
119894 ) minus Γ (119888119899119894 120579119899119894 )] + 119877119899
119888119894
fl 119877119899
119888119894 1 le 119894 le 119872 minus 1 (41)
nabla1199051198881198990 minus 2ℎ [11989912 minus 12057211198881198990 ] = minus [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)]+ 119877119899
1198880 fl 119877119899
1198880(42)
nabla119905119888119899119872 minus 2ℎ [minus1205722119888119899119872 minus 119899119872minus12]
= minus [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] + 119877119899
119888119872 fl 119877119899
119888119872(43)
119899119894+12 = 119888119899119894+12120575119909Θ119899
119894+12 + 120579119899119894+12120575119909119862119899
119894+12
+ 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12 + 119877119899119906119894+120 le 119894 le 119872 minus 1
(44)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581120575lowast119909119908119899
119894 = minus119888119899119894 nabla119905Θ119899119894
+ 12 (119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)+ [(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )]+ 119877119899
120579119894 fl 119877119899
120579119894 1 le 119894 le 119872 minus 1
(45)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [119908119899
12 minus 12057311205791198990]
= minus1198881198990nabla119905Θ1198990 + 119899
12120575119909Θ11989912
+ [(120582 + Θ1198990) Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)]
+ 1198771198991205790 fl 119877119899
1205790(46)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [minus1205732120579119899119872 minus 119908119899
119872minus12] = minus119888119899119872nabla119905Θ119899119872
+ 119899119872minus12120575119909Θ119899
119872minus12
+ [(120582 + Θ119899119872) Γ (119862119899
119872 Θ119899119872) minus (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)]
+ 119877119899120579119872 fl 119877119899
120579119872
(47)
119908119899119894+12 = 120575119909120579119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1 (48)
and
1198880119894 = 01205790119894 = 01198881119894 = 1205911198771
1198881198941205791119894 = 1205911198771
120579119894(49)
and by (40) we can directly derive the inequality
100381710038171003817100381710038171198881100381710038171003817100381710038172 + 100381710038171003817100381710038171205791100381710038171003817100381710038172 + 100381710038171003817100381710038171205751199091205791100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (50)
To prove our main theorem the following formula will beoften used
[V121199110 + ℎ119872minus1sum119894=1
120575119909V119894119911119894 minus V119872minus12119911119872]= 119872minus1sum
119894=0
V119894+12 (119911119896119894 minus 119911119896
119894+1) (51)
In the following lemma we present discrete Sobolev interpo-lation formulas and the proof can be found in [24]
6 Journal of Mathematics
Lemma 2 Let V and 119911 be two mesh functions en for anypositive constant 120598
V2infin le 120598 1003817100381710038171003817120575119909V10038171003817100381710038172 + (1120598 + 1119871) V2 (52)
1003817100381710038171003817120575119909V1003817100381710038171003817infin le 120598 100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 + 119864119888
1003817100381710038171003817120575119909V10038171003817100381710038172 (53)
Lemma 3
10038171003817100381710038171198881198991003817100381710038171003817infin 1003817100381710038171003817100381712057911989910038171003817100381710038171003817infin le 3119864120 (12059174 + ℎ32) 1 le 119899 le 119896 (54)1003817100381710038171003817100381711988811989911990910038171003817100381710038171003817 10038171003817100381710038171003817120579119899
119909
10038171003817100381710038171003817 le 119864120 (12059132 + ℎ) 1 le 119899 le 119896 minus 1 (55)
Proof From (30) for 0 le 119899 le 119896 minus 1 we have10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (56)
When 120591 le ℎ with the inverse inequality we have
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infin le 2ℎminus1 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 le 2ℎminus11198640 (1205912 + ℎ2)2le 81198640ℎ3 (57)
When ℎ le 120591 by taking 120598 = 12059112 in Lemma 2
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infinle 12059112 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 + (120591minus12 + 119871minus1) 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172le (120591minus12 + 12059112119871minus1) 1198640 (1205912 + ℎ2)2 le 8119864012059172
1 le 119894 le 119872 minus 1 0 le 119899 le 119896 minus 1(58)
The first part of (54) is obtained and the second part and theinequality (55) can be proved similarly
In addition by Lemma 3 there exist constants 1198643 gt 0 and1199040 gt 0 such that when ℎ 120591 le 1199040119888min2 le 119888119899119894 le 2119888max120579min2 le 120579119899
119894 le 2120579max0 le 119894 le 119872 minus 1 1 le 119899 le 119896
(59)
and 1003816100381610038161003816nabla119905Θ1198991198941003816100381610038161003816 1003816100381610038161003816nabla119905119862119899
1198941003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816120575119909119862119899
119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119862119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 le 11986430 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816119888119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120579119899119894+12
10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 119896
10038161003816100381610038161003816Γ (119888119899119894+12 120579119899119894+12)10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 1198961003816100381610038161003816(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899
119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 1198961003816100381610038161003816Γ (119862119899
119894 Θ119899119894 ) minus Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816 le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 119896(60)
32 e Existence and Uniqueness Since the coefficientmatrix in the system (18)-(20) is strictly diagonally dominantthus the system (18)-(20) has a unique solution 119888119896+1
119894 Here wewill discuss the boundedness of 119888119896+1
119894 Multiplying (41)-(43) by ℎ119888119899119894 ℎ1198881198990 2 and ℎ1198881198991198722 respec-
tively we get
(nabla119905119888119899 119888119899) + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162 = [119899121198881198990
+ ℎ119872minus1sum119894=1
120575lowast119909 119899
119894 119888119899119894 + 119899119872minus12119888119899119872] + ℎ119872minus1sum
119894=1
119877119899119888119894119888119899119894 + ℎ2
sdot 11987711989911988801198881198990 + ℎ2119877119899
119888119872119888119899119872minus ℎ12 [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)] 1198881198990+ 119872minus1sum
119894=1
[Γ (119862119899119894 Θn
119894 ) minus Γ (119888119899119894 120579119899119894 )] 119888119899119894
+ 12 [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] 119888119899119872 fl 1198691 + 1198692
+ 1198693 1 le 119899 le 119896
(61)
With (44) (51) (59) (60) and (40) we see that
minus 1198691 = minus119899121198881198990 minus ℎ119872minus1sum
119894=1
120575119909119899119894 119888119899119894 + 119899
Mminus12119888119899119872= ℎ119872minus1sum
119894=0
119899119894+12120575119909119888119899119894+12 = ℎ119872minus1sum
119894=0
[119888119899119894+12120575119909Θ119899119894+12
+ 120579119899119894+12120575119909119862119899
119894+12 + 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119877119899119906119894+12] 120575119909119888119899119894+12 ge 120579min2 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 2119888max
1003817100381710038171003817100381712057511990912057911989910038171003817100381710038171003817sdot 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817 minus 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817minus ℎ119872minus1sum
119894=0
10038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119888119899119894+12
10038161003816100381610038161003816 ge 120579min4 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172minus 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(62)
Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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Journal of Mathematics 5
(1198621198990 + 120590)nabla119905Θ119899
0 minus 11988011989912120575119909Θ119899
12
minus 2120581ℎ [11988211989912 minus 1205731 (Θ119899
0 minus ]1)]= (120582 + Θ119899
0) Γ (1198621198990 Θ119899
0) + 1198771198991205790
(119862119899119872 + 120590)nabla119905Θ119899
119872 minus 119880119899119872minus12120575119909Θ119899
119872minus12
minus 2120581ℎ [1205732 (]2 minus Θ119899119872) minus119882119899
119872minus12]= (120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) + 119877119899120579119872
119882119899119894+12 = 120575119909Θ119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1
(38)
and the initial conditions
1198620119894 = 1198880 (119909119894)
Θ0119894 = 1205790 (119909119894)
1198621119894 = 1198880 (119909119894) + 120591119888119905 (119909119894 0) + 1205911198771
119888119894Θ1
119894 = 1205790 (119909119894) + 120591120579119905 (119909119894 0) + 1205911198771120579119894
(39)
where 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 100381610038161003816100381610038161198771120579119894
10038161003816100381610038161003816 le 1198641 (120591 + ℎ2) 0 le 119894 le 11987210038161003816100381610038161003816119877119899119888119894
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119894
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 1 le 119894 le 119872 minus 110038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119908119894+12
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 0 le 119894 le 119872 minus 1100381610038161003816100381610038161198771198991198880
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899119888119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ2) 100381610038161003816100381610038161198771198991205790
10038161003816100381610038161003816 10038161003816100381610038161003816119877119899120579119872
10038161003816100381610038161003816 le 1198642 (1205912 + ℎ) 1 le 119899 le 119873 minus 1
(40)
Subtracting the system (36) from the system (38) we get theerror equations
nabla119905119888119899119894 minus 120575lowast119909 119899
119894 = minus [Γ (119862119899119894 Θ119899
119894 ) minus Γ (119888119899119894 120579119899119894 )] + 119877119899
119888119894
fl 119877119899
119888119894 1 le 119894 le 119872 minus 1 (41)
nabla1199051198881198990 minus 2ℎ [11989912 minus 12057211198881198990 ] = minus [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)]+ 119877119899
1198880 fl 119877119899
1198880(42)
nabla119905119888119899119872 minus 2ℎ [minus1205722119888119899119872 minus 119899119872minus12]
= minus [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] + 119877119899
119888119872 fl 119877119899
119888119872(43)
119899119894+12 = 119888119899119894+12120575119909Θ119899
119894+12 + 120579119899119894+12120575119909119862119899
119894+12
+ 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12 + 119877119899119906119894+120 le 119894 le 119872 minus 1
(44)
(119888119899119894 + 120590)nabla119905120579119899119894 minus 12 (119906119899
119894minus12120575119909120579119899119894minus12 + 119906119899
119894+12120575119909120579119899119894+12)
minus 120581120575lowast119909119908119899
119894 = minus119888119899119894 nabla119905Θ119899119894
+ 12 (119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)+ [(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )]+ 119877119899
120579119894 fl 119877119899
120579119894 1 le 119894 le 119872 minus 1
(45)
(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 2120581ℎ [119908119899
12 minus 12057311205791198990]
= minus1198881198990nabla119905Θ1198990 + 119899
12120575119909Θ11989912
+ [(120582 + Θ1198990) Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)]
+ 1198771198991205790 fl 119877119899
1205790(46)
(119888119899119872 + 120590)nabla119905120579119899119872 minus 119906119899
119872minus12120575119909120579119899119872minus12
minus 2120581ℎ [minus1205732120579119899119872 minus 119908119899
119872minus12] = minus119888119899119872nabla119905Θ119899119872
+ 119899119872minus12120575119909Θ119899
119872minus12
+ [(120582 + Θ119899119872) Γ (119862119899
119872 Θ119899119872) minus (120582 + 120579119899
119872) Γ (119888119899119872 120579119899119872)]
+ 119877119899120579119872 fl 119877119899
120579119872
(47)
119908119899119894+12 = 120575119909120579119899
119894+12 + 119877119899119908119894+120 le 119894 le 119872 minus 1 1 le 119899 le 119873 minus 1 (48)
and
1198880119894 = 01205790119894 = 01198881119894 = 1205911198771
1198881198941205791119894 = 1205911198771
120579119894(49)
and by (40) we can directly derive the inequality
100381710038171003817100381710038171198881100381710038171003817100381710038172 + 100381710038171003817100381710038171205791100381710038171003817100381710038172 + 100381710038171003817100381710038171205751199091205791100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (50)
To prove our main theorem the following formula will beoften used
[V121199110 + ℎ119872minus1sum119894=1
120575119909V119894119911119894 minus V119872minus12119911119872]= 119872minus1sum
119894=0
V119894+12 (119911119896119894 minus 119911119896
119894+1) (51)
In the following lemma we present discrete Sobolev interpo-lation formulas and the proof can be found in [24]
6 Journal of Mathematics
Lemma 2 Let V and 119911 be two mesh functions en for anypositive constant 120598
V2infin le 120598 1003817100381710038171003817120575119909V10038171003817100381710038172 + (1120598 + 1119871) V2 (52)
1003817100381710038171003817120575119909V1003817100381710038171003817infin le 120598 100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 + 119864119888
1003817100381710038171003817120575119909V10038171003817100381710038172 (53)
Lemma 3
10038171003817100381710038171198881198991003817100381710038171003817infin 1003817100381710038171003817100381712057911989910038171003817100381710038171003817infin le 3119864120 (12059174 + ℎ32) 1 le 119899 le 119896 (54)1003817100381710038171003817100381711988811989911990910038171003817100381710038171003817 10038171003817100381710038171003817120579119899
119909
10038171003817100381710038171003817 le 119864120 (12059132 + ℎ) 1 le 119899 le 119896 minus 1 (55)
Proof From (30) for 0 le 119899 le 119896 minus 1 we have10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (56)
When 120591 le ℎ with the inverse inequality we have
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infin le 2ℎminus1 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 le 2ℎminus11198640 (1205912 + ℎ2)2le 81198640ℎ3 (57)
When ℎ le 120591 by taking 120598 = 12059112 in Lemma 2
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infinle 12059112 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 + (120591minus12 + 119871minus1) 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172le (120591minus12 + 12059112119871minus1) 1198640 (1205912 + ℎ2)2 le 8119864012059172
1 le 119894 le 119872 minus 1 0 le 119899 le 119896 minus 1(58)
The first part of (54) is obtained and the second part and theinequality (55) can be proved similarly
In addition by Lemma 3 there exist constants 1198643 gt 0 and1199040 gt 0 such that when ℎ 120591 le 1199040119888min2 le 119888119899119894 le 2119888max120579min2 le 120579119899
119894 le 2120579max0 le 119894 le 119872 minus 1 1 le 119899 le 119896
(59)
and 1003816100381610038161003816nabla119905Θ1198991198941003816100381610038161003816 1003816100381610038161003816nabla119905119862119899
1198941003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816120575119909119862119899
119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119862119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 le 11986430 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816119888119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120579119899119894+12
10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 119896
10038161003816100381610038161003816Γ (119888119899119894+12 120579119899119894+12)10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 1198961003816100381610038161003816(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899
119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 1198961003816100381610038161003816Γ (119862119899
119894 Θ119899119894 ) minus Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816 le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 119896(60)
32 e Existence and Uniqueness Since the coefficientmatrix in the system (18)-(20) is strictly diagonally dominantthus the system (18)-(20) has a unique solution 119888119896+1
119894 Here wewill discuss the boundedness of 119888119896+1
119894 Multiplying (41)-(43) by ℎ119888119899119894 ℎ1198881198990 2 and ℎ1198881198991198722 respec-
tively we get
(nabla119905119888119899 119888119899) + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162 = [119899121198881198990
+ ℎ119872minus1sum119894=1
120575lowast119909 119899
119894 119888119899119894 + 119899119872minus12119888119899119872] + ℎ119872minus1sum
119894=1
119877119899119888119894119888119899119894 + ℎ2
sdot 11987711989911988801198881198990 + ℎ2119877119899
119888119872119888119899119872minus ℎ12 [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)] 1198881198990+ 119872minus1sum
119894=1
[Γ (119862119899119894 Θn
119894 ) minus Γ (119888119899119894 120579119899119894 )] 119888119899119894
+ 12 [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] 119888119899119872 fl 1198691 + 1198692
+ 1198693 1 le 119899 le 119896
(61)
With (44) (51) (59) (60) and (40) we see that
minus 1198691 = minus119899121198881198990 minus ℎ119872minus1sum
119894=1
120575119909119899119894 119888119899119894 + 119899
Mminus12119888119899119872= ℎ119872minus1sum
119894=0
119899119894+12120575119909119888119899119894+12 = ℎ119872minus1sum
119894=0
[119888119899119894+12120575119909Θ119899119894+12
+ 120579119899119894+12120575119909119862119899
119894+12 + 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119877119899119906119894+12] 120575119909119888119899119894+12 ge 120579min2 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 2119888max
1003817100381710038171003817100381712057511990912057911989910038171003817100381710038171003817sdot 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817 minus 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817minus ℎ119872minus1sum
119894=0
10038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119888119899119894+12
10038161003816100381610038161003816 ge 120579min4 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172minus 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(62)
Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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6 Journal of Mathematics
Lemma 2 Let V and 119911 be two mesh functions en for anypositive constant 120598
V2infin le 120598 1003817100381710038171003817120575119909V10038171003817100381710038172 + (1120598 + 1119871) V2 (52)
1003817100381710038171003817120575119909V1003817100381710038171003817infin le 120598 100381710038171003817100381710038171205752
119909V10038171003817100381710038171003817 + 119864119888
1003817100381710038171003817120575119909V10038171003817100381710038172 (53)
Lemma 3
10038171003817100381710038171198881198991003817100381710038171003817infin 1003817100381710038171003817100381712057911989910038171003817100381710038171003817infin le 3119864120 (12059174 + ℎ32) 1 le 119899 le 119896 (54)1003817100381710038171003817100381711988811989911990910038171003817100381710038171003817 10038171003817100381710038171003817120579119899
119909
10038171003817100381710038171003817 le 119864120 (12059132 + ℎ) 1 le 119899 le 119896 minus 1 (55)
Proof From (30) for 0 le 119899 le 119896 minus 1 we have10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 120591 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 le 1198640 (1205912 + ℎ2)2 (56)
When 120591 le ℎ with the inverse inequality we have
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infin le 2ℎminus1 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 le 2ℎminus11198640 (1205912 + ℎ2)2le 81198640ℎ3 (57)
When ℎ le 120591 by taking 120598 = 12059112 in Lemma 2
10038161003816100381610038161003816119888119899+1119894
100381610038161003816100381610038162 le 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172infinle 12059112 10038171003817100381710038171003817120575119909119888119899+1100381710038171003817100381710038172 + (120591minus12 + 119871minus1) 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172le (120591minus12 + 12059112119871minus1) 1198640 (1205912 + ℎ2)2 le 8119864012059172
1 le 119894 le 119872 minus 1 0 le 119899 le 119896 minus 1(58)
The first part of (54) is obtained and the second part and theinequality (55) can be proved similarly
In addition by Lemma 3 there exist constants 1198643 gt 0 and1199040 gt 0 such that when ℎ 120591 le 1199040119888min2 le 119888119899119894 le 2119888max120579min2 le 120579119899
119894 le 2120579max0 le 119894 le 119872 minus 1 1 le 119899 le 119896
(59)
and 1003816100381610038161003816nabla119905Θ1198991198941003816100381610038161003816 1003816100381610038161003816nabla119905119862119899
1198941003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816120575119909119862119899
119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119862119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909Θ119899119894+12
10038161003816100381610038161003816 le 11986430 le 119894 le 119872 minus 1 1 le 119899 le 11989610038161003816100381610038161003816119888119899119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120579119899119894+12
10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 119896
10038161003816100381610038161003816Γ (119888119899119894+12 120579119899119894+12)10038161003816100381610038161003816 le 1198643 0 le 119894 le 119872 minus 1 1 le 119899 le 1198961003816100381610038161003816(120582 + Θ119899
119894 ) Γ (119862119899119894 Θ119899
119894 ) minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899
119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 1198961003816100381610038161003816Γ (119862119899
119894 Θ119899119894 ) minus Γ (119888119899119894 120579119899
119894 )1003816100381610038161003816 le 1198643 (1003816100381610038161003816119888119899119894 1003816100381610038161003816 + 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816) 0 le 119894 le 119872 1 le 119899 le 119896(60)
32 e Existence and Uniqueness Since the coefficientmatrix in the system (18)-(20) is strictly diagonally dominantthus the system (18)-(20) has a unique solution 119888119896+1
119894 Here wewill discuss the boundedness of 119888119896+1
119894 Multiplying (41)-(43) by ℎ119888119899119894 ℎ1198881198990 2 and ℎ1198881198991198722 respec-
tively we get
(nabla119905119888119899 119888119899) + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162 = [119899121198881198990
+ ℎ119872minus1sum119894=1
120575lowast119909 119899
119894 119888119899119894 + 119899119872minus12119888119899119872] + ℎ119872minus1sum
119894=1
119877119899119888119894119888119899119894 + ℎ2
sdot 11987711989911988801198881198990 + ℎ2119877119899
119888119872119888119899119872minus ℎ12 [Γ (119862119899
0 Θ1198990) minus Γ (1198881198990 120579119899
0)] 1198881198990+ 119872minus1sum
119894=1
[Γ (119862119899119894 Θn
119894 ) minus Γ (119888119899119894 120579119899119894 )] 119888119899119894
+ 12 [Γ (119862119899119872 Θ119899
119872) minus Γ (119888119899119872 120579119899119872)] 119888119899119872 fl 1198691 + 1198692
+ 1198693 1 le 119899 le 119896
(61)
With (44) (51) (59) (60) and (40) we see that
minus 1198691 = minus119899121198881198990 minus ℎ119872minus1sum
119894=1
120575119909119899119894 119888119899119894 + 119899
Mminus12119888119899119872= ℎ119872minus1sum
119894=0
119899119894+12120575119909119888119899119894+12 = ℎ119872minus1sum
119894=0
[119888119899119894+12120575119909Θ119899119894+12
+ 120579119899119894+12120575119909119862119899
119894+12 + 119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119877119899119906119894+12] 120575119909119888119899119894+12 ge 120579min2 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 2119888max
1003817100381710038171003817100381712057511990912057911989910038171003817100381710038171003817sdot 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817 minus 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057511990911988811989910038171003817100381710038171003817minus ℎ119872minus1sum
119894=0
10038161003816100381610038161003816119877119899119906119894+12
10038161003816100381610038161003816 10038161003816100381610038161003816120575119909119888119899119894+12
10038161003816100381610038161003816 ge 120579min4 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172minus 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(62)
Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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Journal of Mathematics 7
and by using (60) again we have
100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381711988811989910038171003817100381710038171003817 (63)
and with (52)
100381610038161003816100381611986921003816100381610038161003816 le 12ℎ119872minus1sum119894=1
[(119877119899119888119894)2 + (119888119899119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991198880
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161198881198990 100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899119888119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 10038171003817100381710038171003817119888119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 11986421198712 (1205912 + ℎ2)2+ 1198642ℎ2 (1205912 + ℎ2)2
le 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(64)
Substituting the last three equations into (61) results in
12nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 1205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(65)
where we have noted (nabla119905119888119899 119888119899) = (14120591)(119888119899+12 minus 119888119899minus12) =(12)nabla1199051198881198992 Moreover by the assumption of the induction
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 4120591(120579min8 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 + 1205721
1003816100381610038161003816100381610038161198880 1003816100381610038161003816100381610038162 + 1205722
1003816100381610038161003816100381610038161198881198721003816100381610038161003816100381610038162)le 10038171003817100381710038171003817119888119896minus1100381710038171003817100381710038172+ 4119864119888120591 (10038171003817100381710038171003817120575119909120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 1003817100381710038171003817100381710038171198881003817100381710038171003817100381710038172)+ 4120591119864119888 (1205912 + ℎ2)2
(66)
Since we have the fact that 1198882 le (12)(119888119896+12 + 119888119896minus12)thus
(1 minus 2120591119864119888) 10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min2 1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172le (1198640 + 4120591119864119888 + 41198641198881198640 + 61205911198641198881198640) (1205912 + ℎ2)2 (67)
When 120591119864119888 lt 14 we can get the inequality as
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172 + 120591120579min1003817100381710038171003817100381710038171205751199091198881003817100381710038171003817100381710038172 le 1198644 (1205912 + ℎ2)2 (68)
Since 1198644 are independent of 119896 by (13) when ℎ and 120591 are smallenough
119888119896+1119894 ge 0 0 le 119894 le 119872 (69)
Now we try to prove our main theorem By noting (44)(60) (40) and Lemma 3
10038171003817100381710038171003817119899100381710038171003817100381710038172 = 119872minus1sum119894=0
ℎ [119888119899119894+12120575119909120579119899119894+12 + 120579119899
119894+12120575119909119888119899119894+12
+ 119888119899119894+12120575119909Θ119899119894+12 + 120579119899
119894+12120575119909119862119899119894+12 + 119877119899
119906119894+12]2le 5119872minus1sum
119894=0
ℎ (41198882max10038161003816100381610038161003816120575119909120579119899
119894+12
100381610038161003816100381610038162 + 41205792max
10038161003816100381610038161003816120575119909119888119899119894+12
100381610038161003816100381610038162+ 1198642
3
10038161003816100381610038161003816119888119899119894+12
100381610038161003816100381610038162 + 11986423
10038161003816100381610038161003816120579119899119894+12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899119906119894+12
100381610038161003816100381610038162)le 201198882max
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 201205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 511986423 (100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 511986422119871 (1205912 + ℎ2)2 1 le 119899 le 119896
(70)
We can see that when 120591 le ℎ the assumption of induction and(68) show that
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 401205792max
10038171003817100381710038171003817119888119896+1100381710038171003817100381710038172ℎ2+ 119864119888ℎ4
le 1601205792max1198644ℎ2 + 119864119888ℎ4
(71)
and when ℎ le 120591 by (68)1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 201205792
max1198644 (1205912 + ℎ2)2120591120579min
+ 1198641198881205914
le 801205792max120579min
11986441205913 + 1198641198881205914(72)
which means there exists an 1198645 independent of 119896 such that100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (12059132 + ℎ) (73)
Multiplying the error equation (41) by 120575lowast119909119906
119894 leads to
100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le
1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
nabla119905119888119896119894 120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816 +1003816100381610038161003816100381610038161003816100381610038161003816119872minus1sum119894=1
119877119888119894120575lowast119909119906
119894
1003816100381610038161003816100381610038161003816100381610038161003816le 4 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 81198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 34 100381710038171003817100381710038171003817120575lowast
119909 1003817100381710038171003817100381710038172 + 411986422119871 (1205912 + ℎ2)2
(74)
that is 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 le 16 10038171003817100381710038171003817nabla119905119888119896100381710038171003817100381710038172 + 321198642
3 (10038171003817100381710038171003817120579119896100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119896100381710038171003817100381710038172)+ 161198642
2119871 (1205912 + ℎ2)2 (75)
We can see that when 120591 le ℎ100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin le ℎminus12 100381710038171003817100381710038171003817100381710038171003817100381710038171003817 le 1198645 (ℎ + ℎ12) (76)
8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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8 Journal of Mathematics
and when ℎ le 120591 with Lemma 2
1003817100381710038171003817100381710038171003817100381710038171003817100381710038172infin le 100381710038171003817100381710038171003817120575lowast119909 1003817100381710038171003817100381710038172 + (1 + 119871minus1) 1003817100381710038171003817100381710038171003817100381710038171003817100381710038172 le 11986461205912 (77)
where 1198646 is independent of 119896 Then there exists 1199040 gt 0 whenℎ 120591 le 1199040 100381710038171003817100381710038171003817119906100381710038171003817100381710038171003817infin = max1le119894le119872
100381610038161003816100381610038161003816119906119894minus12
100381610038161003816100381610038161003816 le 21198643 (78)
With a time step condition 120591 le 119864119888ℎ we can see that thecoefficient matrix of the system (21)-(23) is strictly diagonallydominant Thus this system has a unique solution 120579119896+1
119894
33 e Optimal Error Estimate We have proved the exis-tence and uniqueness of the solution to the system and havederived the estimate (65) for 119888119899+1 In this part we try to derivean estimate for 120579119899+1
Multiplying (45)-(47) by ℎ120579119899119894 ℎ120579119899
02 and ℎ1205791198991198722 respec-
tively we try to estimate each term below
ℎ[12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] minus ℎ2 [11990611989912 (120575119909120579119899
12) 1205791198990
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872] minus 120581[119908119899
121205791198990
+ ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894+12120579119899119894 minus 119908119899
119872minus12120579119899119872] + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 = minusℎ[121198881198990 (nabla119905Θ1198990) 120579119899
0
+ 119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 120579119899
119894 + 12119888119899119872 (nabla119905Θ119899119872) 120579119899
119872]+ ℎ2 [119899
12 (120575119909Θ11989912) 120579119899
0
+ 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 120579119899119894
+ 119899119872minus12 (120575119909Θ119899
119872minus12) 120579119899119872] + ℎ2 [(120582 + Θ119899
0)sdot Γ (119862119899
0 Θ1198990) minus (120582 + 120579119899
0) Γ (1198881198990 1205791198990)] 120579119899
0
+ ℎ119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 ) minus (120582 + 120579119899
119894 ) Γ (119888119899119894 120579119899119894 )]
sdot 120579119899119894 + ℎ2 [(120582 + Θ119899
119872) Γ (119862119899119872 Θ119899
119872) minus (120582 + 120579119899119872)
sdot Γ (119888119899119872 120579119899119872)] 120579119899
119872 + [ℎ119872minus1sum119894=1
119877119899120579119894120579119899
119894 + ℎ21198771198991205790120579119899
0 + ℎ2sdot 119877119899
120579119872120579119899119872] fl 1198693 + 1198694 + 1198695 + 1198696 1 le 119899 le 119896
(79)According to Lemma 2 (40) (51) and (60) three terms
on the left can be bounded by
ℎ [12 (1198881198990 + 120590) (nabla1199051205791198990) 120579119899
0 + 119872minus1sum119894=1
(119888119899119894 + 120590) (nabla119905120579119899119894 ) 120579119899
119894
+ 12 (119888119899119872 + 120590) (nabla119905120579119899119872) 120579119899
119872] = ℎ2 12 (1198881198990 + 120590)sdot nabla119905 [(120579119899
0)2] + 119872minus1sum119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 12 (119888119899119872
+ 120590)nabla119905 [(120579119899119872)2]
1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [119906119899
12 (12057511990912057911989912) 120579119899
0
+ 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 120579119899119894
+ 119906119899119872minus12 (120575119909120579119899
119872minus12) 120579119899119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 1003816100381610038161003816100381610038161003816100381610038161003816minusℎ119872minus1sum119894=0
119906119899119894+12120575119909120579119899
119894+12120579119899119894+12
1003816100381610038161003816100381610038161003816100381610038161003816le 21198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
120575119909120579119899119894+12120579119899
119894+12
1003816100381610038161003816100381610038161003816100381610038161003816 le1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172
(80)
and
minus 120581[11990811989912120579119899
0 + ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 120579119899119894 minus 119908119899
119872minus12120579119899119872]
= 120581ℎ119872minus1sum119894=0
(120575119909120579119899119894+12)2 + 120581ℎ119872minus1sum
119894=0
119877119908119894+12120575119909120579119899119894+12
ge 120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888ℎ119872minus1sum119894=0
10038161003816100381610038161003816119877119899119908119894+12
100381610038161003816100381610038162 minus 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172ge 31205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(81)
By (70) for those terms in the right hand side we obtain100381610038161003816100381611986931003816100381610038161003816 le 1198643 (10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) 100381610038161003816100381611986941003816100381610038161003816 le ℎ1198643 (10038161003816100381610038161003816119899
12
10038161003816100381610038161003816 sdot 100381610038161003816100381610038161205791198990
10038161003816100381610038161003816 + 119872minus1sum119894=1
(10038161003816100381610038161003816119899119894minus12
10038161003816100381610038161003816 + 10038161003816100381610038161003816119899119894+12
10038161003816100381610038161003816) 10038161003816100381610038161003816120579119899119894
10038161003816100381610038161003816
Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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Journal of Mathematics 9
+ 10038161003816100381610038161003816119899119872minus12
10038161003816100381610038161003816 sdot 10038161003816100381610038161003816120579119899119872
10038161003816100381610038161003816) le 21198643 (10038171003817100381710038171003817120579119899100381710038171003817100381710038172
+ 119872minus1sum119894=0
ℎ 10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162) le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 + 100381710038171003817100381711988811989910038171003817100381710038172) + 1198642 (1205912 + ℎ2)2
100381610038161003816100381611986951003816100381610038161003816 le 1198643 (10038171003817100381710038171198881198991003817100381710038171003817 + 1003817100381710038171003817100381712057911989910038171003817100381710038171003817) 1003817100381710038171003817100381712057911989910038171003817100381710038171003817 100381610038161003816100381611986961003816100381610038161003816 le 12ℎ
119872minus1sum119894=1
[(119877119899120579119894)2 + (120579119899
119894 )2] + ℎ2
2 100381610038161003816100381610038161198771198991205790
100381610038161003816100381610038162 + 12 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162
+ ℎ2
2 10038161003816100381610038161003816119877119899120579119872
100381610038161003816100381610038162 + 12 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 10038171003817100381710038171003817120579119899100381710038171003817100381710038172infin + 119864119888
10038171003817100381710038171003817120579119899100381710038171003817100381710038172+ 11986421198712 (1205912 + ℎ2)2 + 1198642ℎ2 (1205912 + ℎ)2 le 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2
(82)
Taking the last six equations into (79) we obtain
ℎ2 12 (1198881198990 + 120590)nabla119905 [(1205791198990)2] + 119872minus1sum
119894=1
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2]
+ 12 (119888119899119872 + 120590)nabla119905 [(120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)+ 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(83)
Besides we introduce a notation as 120579lowast119894 = (12)[(120579119899+1
119894 )2 +(120579119899minus1119894 )2] and adding the first three equations into (36) byℎ120579lowast119894 2 ℎ120579lowast
0 4 and ℎ120579lowast1198724 respectively we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]minus 12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]
+ 12057212 (1198881198990 minus 1205831) 120579lowast0 + 12057222 (119888119899119872 minus 1205832) 120579lowast
119872
= minusℎ2 [12120579lowast0 Γ (1198881198990 120579119899
0) + 119872minus1sum119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)] 1 le 119899 le 119896
(84)
We now estimate the terms in (84) By (51) we denote
100381610038161003816100381611986971003816100381610038161003816 fl 1003816100381610038161003816100381610038161003816100381610038161003816minus12 [119906119899
12120579lowast0 + ℎ119872minus1sum
119894=1
120575lowast119909119906119899
119894+12120579lowast119894 minus 119906119899
119872minus12120579lowast119872]1003816100381610038161003816100381610038161003816100381610038161003816
= 100381610038161003816100381610038161003816100381610038161003816100381612
119872minus1sum119894=0
119906119899119894+12 (120579lowast
119894+1 minus 120579lowast119894 )1003816100381610038161003816100381610038161003816100381610038161003816
le 10038161003816100381610038161003816100381610038161003816100381610038161198643ℎ119872minus1sum119894=0
(120579119899+1119894+12120575119909120579119899+1
119894+12 + 120579119899minus1119894+12120575119909120579119899minus1
119894+12)1003816100381610038161003816100381610038161003816100381610038161003816le 11986432 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)
(85)
Using (60) again we get1003816100381610038161003816100381610038161003816100381610038161003816minusℎ2 [12120579lowast
0 Γ (1198881198990 1205791198990) + 119872minus1sum
119894=1
120579lowast119894 Γ (119888119899119894 120579119899
119894 )+ 12120579lowast
119872Γ (119888119899119872 120579119899119872)]1003816100381610038161003816100381610038161003816100381610038161003816 le
11986434 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) (86)
and with Lemma 2 we have
ℎ2 [12 (nabla1199051198881198990 ) 120579lowast0 + 119872minus1sum
119894=1
(nabla119905119888119899119894 ) 120579lowast119894 + 12 (nabla119905119888119899119872) 120579lowast
119872]le 120572112058312 120579lowast
0 + 120572212058322 120579lowast119872
+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)le 12057211205831 + 120572212058322 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172infin + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172infin)+ 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
le 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(87)
Moreover by noting the fact that
(119888119899119894 + 120590)nabla119905 [(120579119899119894 )2] + 120579lowast
119894 nabla119905119888119899119894 = nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2] (88)
adding (87) in (84) and using Lemma 2 again we further get
ℎ2 12nabla119905 [(1198881198990 + 120590) (1205791198990)2] + 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) (120579119899119894 )2]
+ 12nabla119905 [(119888119899119872 + 120590) (120579119899119872)2] + 1205814 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162+ 1205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162 le 4011986431205792max
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2 1 le 119899 le 119896
(89)
10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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10 Journal of Mathematics
Multiplying the last equation with 1199041 = 120579min32011986431205792max and
adding it into equation (65) we get
nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120579min8 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + [21205721
100381610038161003816100381610038161198881198990 100381610038161003816100381610038162 + 21205722
10038161003816100381610038161003816119888119899119872100381610038161003816100381610038162+ 11990411205811205731
100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 11990411205811205732
10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]+ 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]+ 12058111990414 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 le 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172) + 119864119888 (1205912
+ ℎ2)2 1 le 119899 le 119896
(90)
Finally we estimate 120575119909120579119899 Multiplying the error equa-tion (45) by minusℎ1205752
119909120579119899(119888119899119894 + 120590) and summing up the resultingequations for 119894 = 1 2 119872 minus 1 we haveminus ℎ119872minus1sum
119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 + 120581119888119899119894 + 120590ℎ
119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894 = ℎ119888119899119894 + 120590sdot 119872minus1sum
119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12) 1205752119909120579119899
119894
minus ℎ119888119899119894 + 120590119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894 1 le 119899 le 119896
(91)
For the first term we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894
= minus ℎ2120591119872minus1sum119894=1
120579119899+1119894 1205752
119909120579119899119894 + ℎ2120591
119872minus1sum119894=1
120579119899minus1119894 1205752
119909120579119899119894
= minus 12120591119872minus1sum119894=1
120579119899+1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
+ 12120591119872minus1sum119894=1
120579119899minus1119894 (120575119909120579119899
119894+12 minus 120575119909120579119899119894minus12)
(92)
By (51) we have
minus ℎ119872minus1sum119894=1
(nabla119905120579119899119894 ) 1205752
119909120579119899119894 = minus 12120591 (120579119899+1
119872 120575119909120579119899119872minus12
minus 120579119899+11 120575119909120579119899
12 minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899+1
119894+1 minus 120579119899+1119894 ))
+ 12120591 (120579119899minus1119872 120575119909120579119899
119872minus12 minus 120579119899minus11 120575119909120579119899
12
minus 119872minus1sum119894=1
120575119909120579119899119894+12 (120579119899minus1
119894+1 minus 120579119899minus1119894+1 )) = minusnabla119905120579119899
119872120575119909120579119899119872minus12
+ nabla1199051205791198991120575119909120579119899
12 + 14120591 (ℎ119872minus1sum119894=1
(120575119909120579119899+1119894+12)2
minus ℎ119872minus1sum119894=1
(120575119909120579119899minus1119894+12)2) = 12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus nabla119905120579119899
119872120575119909120579119899119872minus12 + nabla119905120579119899
012057511990912057911989912
(93)
For the second term we have
120581119888119899119894 + 120590ℎ119872minus1sum119894=1
120575lowast119909119908119899
119894 1205752119909120579119899
119894
ge 120581119888max + 120590ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162
+ 120581119888max + 120590ℎ119872minus1sum119894=1
120575lowast119909119877119899
1199081198941205752119909120579119899
119894
ge 1205812 (119888max + 120590)ℎ119872minus1sum119894=1
100381610038161003816100381610038161205752119909120579119899
119894
100381610038161003816100381610038162 minus 119864119888 (1205912 + ℎ2)2
(94)
where we noted the fact that |120575lowast119909119877119899
119908119894| le 119864119888(1205912+ℎ2) From (94)we can get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205812 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + nabla119905120579119899
012057511990912057911989912
minus nabla119905120579119899119872120575119909120579119899
119872minus12 le ℎ119888119899119894 + 120590119872minus1sum119894=1
119888119899119894 (nabla119905Θ119899119894 ) 1205752
119909120579119899119894
minus ℎ2 (119888119899119894 + 120590)119872minus1sum119894=1
(119899119894minus12120575119909Θ119899
119894minus12 + 119899119894+12120575119909Θ119899
119894+12)sdot 1205752
119909120579119899119894 minus ℎ119888119899119894 + 120590
119872minus1sum119894=1
[(120582 + Θ119899119894 ) Γ (119862119899
119894 Θ119899119894 )
minus (120582 + 120579119899119894 ) Γ (119888119899119894 120579119899
119894 ) + 119877119899120579119894] 1205752
119909120579119899119894 minus ℎ2 (119888119899119894 + 120590)
Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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Journal of Mathematics 11
sdot 119872minus1sum119894=1
(119906119899119894minus12120575119909120579119899
119894minus12 + 119906119899119894+12120575119909120579119899
119894+12) 1205752119909120579119899
119894
+ 119864119888 (1205912 + ℎ2)2 fl 1198698 + 1198699 + 11986910 + 11986911 1 le 119899 le 119896(95)
Then we estimate the termnabla1199051205791198990120575119909120579119899
12 minusnabla119905120579119899119872120575119909120579119899
119872minus12 and 1198698to 11986911 respectively From (46) we have
12057511990912057911989912 = 1205731120579119899
0
+ ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990 minus 119906119899
1212057511990912057911989912 minus 119877119899
1205790]minus 119877119899
11990812(96)
A straightforward calculation with Lemma 2 leads to
nabla1199051205791198990120575119909120579119899
12 = 1205731nabla1199051205791198990120579119899
0 + ℎ2120581 [(1198881198990 + 120590)nabla1199051205791198990
minus 11990611989912120575119909120579119899
12 minus 119877119899
1205790] nabla1199051205791198990 minus 119877119899
11990812nabla1199051205791198990 ge 12057312
sdot nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205902120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162 minus ℎ212058111990611989912120575119909120579119899
12nabla1199051205791198990
minus ℎ2120581119877119899
1205790nabla1199051205791198990 minus nabla119905 (119877119899
119908121205791198990) + 119877119899+1
11990812 minus 119877119899119908122120591
sdot 120579119899+10 + 119877119899
11990812 minus 119877119899minus1119908122120591 120579119899minus1
0 ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162)+ ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
0
100381610038161003816100381610038162 minus ℎ2120581120590 [1003816100381610038161003816100381611990611989912120575119909120579119899
12
100381610038161003816100381610038162 + 10038161003816100381610038161003816119877119899
1205790
100381610038161003816100381610038162]minus nabla119905 (119877119899
119908121205791198990) minus 119864119888 (1205912 + ℎ2) (10038171003817100381710038171003817120579119899+110038171003817100381710038171003817infin
+ 10038171003817100381710038171003817120579119899minus110038171003817100381710038171003817infin) ge 12057312 nabla119905 (100381610038161003816100381610038161205791198990
100381610038161003816100381610038162) + ℎ1205904120581 10038161003816100381610038161003816nabla1199051205791198990
100381610038161003816100381610038162minus nabla119905 (119877119899
119908121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172) minus 119864119888 (1205912 + ℎ2)2
(97)
where we noted the fact that (119877119899+111990812minus119877119899
11990812)2120591 le 119864119888(1205912+ℎ2)and (119877119899
11990812 minus 119877119899minus111990812)2120591 le 119864119888(1205912 + ℎ2) Similarly we estimateminusnabla119905120579119899
119872120575119909120579119899119872minus12 From (47) we have
minus 120575119909120579119899119872minus12
= 1205732120579119899119872
+ ℎ2120581 [(119888119899119872 + 120590)nabla1199051205791198990 minus 119906119899
119872minus12120575119909120579119899119872minus12 minus 119877119899
120579119872]+ 119877119899
119908119872minus12(98)
such that
minus nabla119905120579119899119872120575119909120579119899
119872minus12 ge 12057322 nabla119905 ((120579119899119872)2) + ℎ1205904120581 10038161003816100381610038161003816nabla119905120579119899
119872
100381610038161003816100381610038162+ nabla119905 (119877119899
119908119872minus121205791198990) minus 21198642
3120581120590 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172minus 501198642
31205792max120581120590 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 minus 119864119888 (1205912 + ℎ2)2 minus 119864119888 (10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172
+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172)
(99)
For those terms in the right hand side with (70) we have
100381610038161003816100381611986981003816100381610038161003816 le 119864119888
10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 (100)
100381610038161003816100381611986991003816100381610038161003816 le 11986432 (119888min + 120590)1003816100381610038161003816100381610038161003816100381610038161003816ℎ
119872minus1sum119894=1
(119899119894minus121205752
119909120579119899119894 + 119899
119894+121205752119909120579119899
119894 )1003816100381610038161003816100381610038161003816100381610038161003816le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752
119909120579119899100381710038171003817100381710038172 + 1611986423 (119888max + 120590)(119888min + 120590)2 120581
sdot ℎ119872minus1sum119894=1
10038161003816100381610038161003816119899119894+12
100381610038161003816100381610038162 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 32011986423 (119888max + 120590) 1205792
max(119888min + 120590)2 12058110038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(101)
1003816100381610038161003816119869101003816100381610038161003816 le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 119864119888 (100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172)
+ 119864119888 (1205912 + ℎ2)2 (102)
1003816100381610038161003816119869111003816100381610038161003816 le 1198643
1003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=1
(120575119909120579119899119894minus121205752
119909120579119899119894minus12
+ 120575119909120579119899119894+121205752
119909120579119899119894+12)1003816100381610038161003816100381610038161003816100381610038161003816
le 21198643
100381610038161003816100381610038161003816100381610038161003816100381610038161003816ℎ119872minus1sum119894=0
[[(radic321198643 (119888max + 120590)120581 120575119909120579119899
119894+12)
sdot (radic 120581321198643 (119888max + 120590)1205752119909120579119899
119894+12)]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816
le 12058116 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172
+ 6411986423 (119888max + 120590)120581 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
(103)
12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
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12 Journal of Mathematics
Plugging the last six equations into (95) we get
12nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 12057312 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 12057322 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + nabla119905 (119877119899119908119872minus12120579119899
119872 minus 11987711989911990812120579119899
0)le 1198647
10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1198648
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 119864119888 (1205912 + ℎ2)2+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172)
(104)
Multiplying the last equation with 0 le 1199042 le 1198649 and adding itinto (90) we have11990422 nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 11990421205814 (119888max + 120590) 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 120579min16 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172
+ 12058111990418 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 120573111990422 nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162)+ 120573211990422 nabla119905 (10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162) + 1199042nabla119905 (minus11987711989911990812120579119899
0
+ 119877119899119908119872minus12120579119899
119872) + 1199041ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162] + 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162]]le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172)+ 119864119888 (10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172 + 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172+ 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172) + 119864119888 (1205912 + ℎ2)2
(105)
Letting 119904 = min(11990422 11990421205812(119888max + 120590) 120579min8 12058111990414 11199041 119904212057312 119904212057322) we get119904nabla119905
10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 1199042 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 1199042 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172
+ 119904nabla119905100381710038171003817100381711988811989910038171003817100381710038172 + 119904nabla119905 (10038161003816100381610038161003816120579119899
0
100381610038161003816100381610038162) + 119904nabla119905 (10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162)+ 119904nabla119905 (minus119877119899
119908121205791198990 + 119877119899
119908119872minus12120579119899119872)
+ 119904ℎ2 [12nabla119905 [(1198881198990 + 120590) 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162]+ 119872minus1sum
119894=1
nabla119905 [(119888119899119894 + 120590) 10038161003816100381610038161003816120579119899119894
100381610038161003816100381610038162]+ 12nabla119905 [(119888119899119872 + 120590) 10038161003816100381610038161003816120579119899
119872
100381610038161003816100381610038162]] le 119864119888 (10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172+ 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381710038171003817100381711988811989910038171003817100381710038172
+ 10038171003817100381710038171003817119888119899minus1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899+1100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120579119899minus1100381710038171003817100381710038172)+ 119864119888 (10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899minus10
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 10038161003816100381610038161003816120579119899minus1
119872
100381610038161003816100381610038162) + 119864119888 (1205912 + ℎ2)2 (106)
Letting
119865119899+1 = 119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 100381610038161003816100381610038161205791198990
100381610038161003816100381610038162 + 10038161003816100381610038161003816120579119899119872
100381610038161003816100381610038162+ 119904ℎ2 (12 (119888119899+1
0 + 120590) 10038161003816100381610038161003816120579119899+10
100381610038161003816100381610038162
+ 119872minus1sum119894=1
((119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162)+ 12 (119888119899+1
119872 + 120590) 10038161003816100381610038161003816120579119899+1119872
100381610038161003816100381610038162)
(107)
we have
119865119899+1 minus 119865119899minus1 + 119904120591 (10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 2120591119864119888 (119865119899+1 + 119865119899 + 119865119899minus1) + 2120591119864119888 (1205912 + ℎ2)2 (108)
fromwhich we can find 120572 = (minus120591119864119888+radic1 minus 3(120591119864119888)2)(1minus2120591119864119888)and 120582 = (120591119864119888 + radic1 minus 3(120591119864119888)2)(1 minus 2120591119864119888) such that
119865119899+1 + 120572119865119899
+ 1199041205911 minus 2120591119864119888
(10038171003817100381710038171003817120575119909119888119899100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119899100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 120582 (119865119899 + 120572119865119899minus1) + 21205911198641198881 minus 2120591119864119888
(1205912 + ℎ2)2 (109)
When 120591119864119888 lt 14119865119899+1 + 120572119865119899 + 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1198651 + 1198650 + (1205912 + ℎ2)2) (110)
and
119904 10038171003817100381710038171003817120575119909120579119899+1100381710038171003817100381710038172 + 119904 10038171003817100381710038171003817119888119899+1100381710038171003817100381710038172 + 119904ℎ2 (12 (119888119899+10 + 120590) 10038161003816100381610038161003816120579119899+1
0
100381610038161003816100381610038162
+ 119872minus1sum119894=1
(119888119899+1119894 + 120590) 10038161003816100381610038161003816120579119899+1
119894
100381610038161003816100381610038162 + 12 (119888119899+1119872 + 120590) 10038161003816100381610038161003816120579119899+1
119872
100381610038161003816100381610038162)+ 119899sum
119898=1
120591119904 (10038171003817100381710038171003817120575119909119888119898100381710038171003817100381710038172 + 10038171003817100381710038171003817120575119909120579119898100381710038171003817100381710038172 + 100381710038171003817100381710038171205752119909120579119899100381710038171003817100381710038172)
le 1198906119864119888T (1205912 + ℎ2)2
(111)
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Journal of Mathematics 13
Table 1 Numerical results of Example 1 with 120591 = ℎ119888119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 2914E-03 1004E-03 0254E-03 199119905 = 050 8163E-03 2178E-03 0527E-03 203119905 = 075 1397E-02 0350E-02 0084E-02 204119905 = 100 2032E-02 0496E-02 0118E-02 205120579119899119905 ℎ = 119871200 ℎ = 119871400 ℎ = 119871800 Order 119903119905 = 025 9509E-03 3795E-03 0947E-03 200119905 = 050 3860E-03 1042E-03 0253E-03 203119905 = 075 7115E-02 1799E-02 0439E-02 202119905 = 100 1069E-01 0265E-01 0065E-01 202
Since 119864119888 is independent of 1198640 with
1198640 = 1198906119864119888T119904 (112)
we find that (30) holds for 119896 = 119899The induction and the proofof the theorem are completed
4 Numerical Examples
We now numerically evaluate the performance of the pro-posed leap-frog scheme
Example 1 First we test the accuracy of our algorithm in anartificial example which is taken from [23] The system is
119862119905 + (119906119892119862)119909 = minusΓ + 119891119862 (119909 119905) (119862 + 120590) 119879119905 minus (120581119879119909)119909 + 119906119892119862119909119879119909
= [120582 + 120590119879] Γ + 119891119879 (119909 119905) (113)
with the boundary conditions (8)-(12) where 119891119862 119891119879 120583119894 and]119894 are coefficients decided by the exact solution
119862 (119909 119905) = 119890minus0721199051199092 (1 minus 119909)2 + 119886 (1 minus 119909) + 119887119909119879 (119909 119905) = 50119890minus0721199051199092 (1 minus 119909)2 + 119888 (1 minus 119909) + 119889119909 (114)
with 119886 119887 119888 119889 being constantsWe apply the uncoupled leap-frog finite difference
method to solve the artificial example We choose T = 1and 119871 = 1 Since the proposed scheme is of the second orderin both spatial and temporal directions we take 120591 = ℎ suchthat the error bound is proportional to ℎ2 We present the1198712-norm errors and the order of convergence ℎ119903 in Table 1with ℎ = 119871200 119871400 119871800 at different time level Wecan see clearly from Table 1 that the 1198712-norm errors forboth components are proportional to ℎ2 which confirms ourtheoretical analysis
Example 2 In the second example we discuss a typicalclothing assembly in the textile industry [2 4 25] The
Table 2 Physical parameters for batting materials
parameter polyester unit120588119891 139 times 103 kgmminus3
120588119908 1 times 103 kgmminus3
1205981015840 0993120581119891 1 times 10minus1 Wmminus1Kminus1
120581119908 57 times 10minus1 Wmminus1Kminus1
119862V119891 117 times 106 J mminus1Kminus1
119871 492 times 10minus2 m
clothing assembly consists of three layers in the middleis porous fibrous media and the outside cover is exposedto a cold environment with fixed temperature and relativehumidity while the inside cover is exposed to a mixture ofair and vapor at higher temperature and relative humidity Inthis paper polyester porous media with laminated or nyloncover materials are tested To compare with the experimentaldata in [12] a water equation is added to equations (1)-(2)
120597120597119905 (120588119908 (1 minus 1205981015840)119882) = 119872119908Γ119888119890 (115)
where 119882 is water content 120588119908 is the density of water 120598 isthe porosity with liquid water content and 1205981015840 is the porositywithout liquid water content We have
120598 = 1205981015840 minus 120588119891120588119908
119882(1 minus 1205981015840) (116)
and the effective heat conductivity is defined by
120581 = 120598120581119892 + (1 minus 120598) 120581119904 (117)
where 120581119892 is the thermal conductivity of gas and 120581119904 is thethermal conductivity of the fiber-watermixture [2 6 7] givenby
120581119904 = ( 1120588119891
+ 119882120588119908
)( 1120588119891120581119891
+ 119882120588119908120581119908
)minus1 (118)
The values of these physical parameters for polyester mediaare presented in Table 2 Other parameters values can befound in [2 6 7]
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
14 Journal of Mathematics
0 1 2 3 4 505
1
15
2
25
C (m
olm
3 )8 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 50
100
200
300
x (cm)
0
100
200
300
Wat
er co
nten
t (
)
1 2 3 4 50x (cm)
Tem
pera
ture
(∘C)
Figure 1 Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric
The initial conditions for the vapor temperature andwater content are given by
119862 = 65119875119904119886119905 (119879)119877119879 119879 = 25∘119862 119882 = 0 at 119905 = 0 (119)
We apply the uncoupled leap-frog finite differencemethod for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115) Since only the rightside of the water equation includes 119888 and 120579 therefore thewater equation is calculated separately Numerically at eachtime step we first find solution 119888119899+1
119895 120579119899+1119895 by procedure (18)-
(26) and then 119882119899+1119895 can be solved by following nonnormal-
ized discrete formate
120588119891 (1 minus 1205981015840119899119895)120591 (119882119899+1119895 minus119882119899
119895 ) = 119872119908Γ119899cej (120)
Then we evaluate the parameters explicitly in (18)-(26) basedon 119882119899+1
119895 Here all numerical results are obtained by takingthe time step size 120591 = 20119904 and spatial mesh size ℎ =119871100 We present numerical results of vapor temperatureand water content at 8 hours and 24 hours respectively for
the porous polyester media assembly with laminated cover inFigure 1 and with nylon cover in Figure 2 The comparisonsbetween numerical results of water content and experimentalmeasurements [12] are given in last two subfigures where theblue lines represent the numerical solution and the red line isgiven by experimental measurement
5 Conclusion
As a subsequent work of [23] we have presented an uncou-pled leap-frog finite differencemethod for the sweat transportsystem in porous textile media which is governed by astrongly coupled nonlinear parabolic system Optimal 1198712
error estimates were presented which imply that the numeri-cal scheme is unconditionally stable Both theoretical analysisand numerical example indicate that the current schemeis second order accurate in both the temporal and spatialdirections Since the scheme is decoupled for the system themethod can be applied efficiently for problems in higher-dimensional space Under certain time-step restrictionsthe analysis can also be extended to the multidimensionalproblems
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Journal of Mathematics 15
C (m
olm
3 )
0 1 2 3 4 50
1
2
38 hours
0 1 2 3 4 50
1
2
324 hours
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5minus20
0
20
40
0 1 2 3 4 5x (cm)
0 1 2 3 4 50
100
200
300
400
x (cm)
Tem
pera
ture
(∘C)
0
100
200
300
400
Wat
er co
nten
t (
)
Figure 2 Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors wish to thank Professors Z Sun and W Sunfor helpful discussions This research was partially supportedby National Natural Science Foundation of China (Nos11601346 11501377) Shenzhen Key Laboratory of AdvancedMachine Learning and Applications Guangdong Key Labo-ratory of Intelligent Information Processing and Interdisci-plinary Innovation Team of Shenzhen University
References
[1] M K Choudhary K C Karki and S V Patankar ldquoMathemat-ical modeling of heat transfer condensation and capillary flowin porous insulation on a cold piperdquo International Journal ofHeat and Mass Transfer vol 47 pp 5629ndash5638 2004
[2] J Fan X Cheng X Wen and W Sun ldquoAn improved modelof heat and moisture transfer with phase change and mobilecondensates in fibrous insulation and comparison with experi-mental resultsrdquo International Journal of Heat andMass Transfervol 47 no 10-11 pp 2343ndash2352 2004
[3] C V le N G ly and R Postle ldquoHeat and Moisture Transferin Textile Assemblies Part I Steaming of Wool Cotton Nylonand Polyester Fabric Bedsrdquo Textile Research Journal vol 65 no4 pp 203ndash212 1995
[4] Y li and Q Zhu ldquoSimultaneous Heat and Moisture Transferwith Moisture Sorption Condensation and Capillary LiquidDiffusion in Porous Textilesrdquo Textile Research Journal vol 73no 6 pp 515ndash524 2003
[5] P W Gibson and M Charmchi ldquoModeling convectiondiffu-sion processes in porous textiles with inclusion of humidity-dependent air permeabilityrdquo International Communications inHeat and Mass Transfer vol 24 no 5 pp 709ndash724 1997
[6] H Huang C Ye and W Sun ldquoMoisture transport in fibrousclothing assembliesrdquo Journal of Engineering Mathematics vol61 no 1 pp 35ndash54 2008
[7] C Ye H Huang J Fan and W Sun ldquoNumerical study of heatand moisture transfer in textile materials by a finite volumemethodrdquo Communications in Computational Physics vol 4 no4 pp 929ndash948 2008
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
16 Journal of Mathematics
[8] Q Zhang B Li and W Sun ldquoHeat and sweat transportthrough clothing assemblies with phase changes condensa-tionevaporation and absorptionrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol467 no 2136 pp 3469ndash3489 2011
[9] Q Zhang ldquoMathematical modeling and numerical study ofcarbonation in porous concretematerialsrdquoAppliedMathematicsand Computation vol 281 pp 16ndash27 2016
[10] C Ye B Li and W Sun ldquoQuasi-steady-state and steady-statemodels for heat and moisture transport in textile assembliesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2122 pp 2875ndash2896 2010
[11] F E Jones Evaporation of Water Lewis Publishers Inc Michi-gan Mich USA 1992
[12] J Fan X Cheng andY S Chen ldquoAn experimental investigationof moisture absoption and condensation in fibrous insulationsunder low temperaturerdquo Experimental ermal and Fluid Sci-ence vol 27 pp 723ndash729 2002
[13] W Dai and G Li ldquoA finite difference scheme for solvingparabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasersrdquoNumericalMethods for Partial Differential Equations vol 22 no6 pp 1396ndash1417 2006
[14] R Eymard J Fuhrmann and K Gartner ldquoA finite volumescheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problemsrdquo Numerische Mathe-matik vol 102 pp 463ndash495 2006
[15] R Eymard TGallouet RHerbin andAMichel ldquoConvergenceof a finite volume scheme for nonlinear degenerate parabolicequationsrdquo Numerische Mathematik vol 92 pp 41ndash82 2002
[16] C V Pao ldquoNumerical analysis of coupled systems of nonlinearparabolic equationsrdquo SIAM Journal on Numerical Analysis vol36 no 2 pp 393ndash416 1999
[17] H J Schroll ldquoConvergence of implicit finite differencemethodsapplied to nonlinear mixed systemsrdquo SIAM Journal on Numeri-cal Analysis vol 33 no 3 pp 997ndash1013 1996
[18] W Sun and G Yuan ldquoStability condition for difference schemesfor parabolic systemsrdquo SIAM Journal onNumerical Analysis vol38 no 2 pp 548ndash555 2000
[19] H Wang ldquoAn optimal-order error esti mate for a familyof ELLAM-MFEM approximations to porous medium flowrdquoSIAM Journal on Numerical Analysis vol 46 no 4 pp 2133ndash2152 2008
[20] Y Yuan ldquoThe upwind finite difference fractional steps methodsfor two-phase compressible flow in porous mediardquo NumericalMethods for Partial Differential Equations vol 19 no 1 pp 67ndash88 2003
[21] B Li W Sun and Y Wang ldquoGlobal existence of weak solutionto the heat and moisture transport system in fibrous porousmediardquo Journal of Differential Equations vol 249 no 10 pp2618ndash2642 2010
[22] B Li and W Sun ldquoGlobal existence of weak solution fornonisothermal multicomponent flow in porous textile mediardquoSIAM Journal onMathematical Analysis vol 42 no 6 pp 3076ndash3102 2010
[23] W Sun and Z Sun ldquoFinite difference methods for a nonlinearand strongly coupled heat and moisture transport system intextile materialsrdquo Numerische Mathematik 2011
[24] A A Samarskii and B B Andreev Finite Difference Methods forElliptic Equation Nauka Moscow Russia 1976
[25] J Fan Z Luo and Y Li ldquoHeat and moisture transfer withsorption and condensation in porous clothing assemblies andnumerical simulationrdquo International Journal of Heat and MassTransfer vol 43 pp 2989ndash3000 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
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