Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 176730, 10 pages http://dx.doi.org/10.1155/2013/176730 Research Article Approximate Solutions of Fisher’s Type Equations with Variable Coefficients A. H. Bhrawy 1,2 and M. A. Alghamdi 1 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt Correspondence should be addressed to A. H. Bhrawy; [email protected] Received 6 September 2013; Accepted 20 September 2013 Academic Editor: Dumitru Baleanu Copyright © 2013 A. H. Bhrawy and M. A. Alghamdi. 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. e spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time- dependent Fisher’s type problems. e spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. e proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. e four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations. 1. Introduction Spectral methods (see, for instance, [1–5]) are powerful tech- niques that we use to numerically solve linear and nonlinear partial differential equations either in their strong or weak forms. What sets spectral methods apart from others like finite difference methods or finite element methods is that to get a spectral method we approximate the solutions by high order orthogonal polynomial expansions. e orthogonal polynomial approximations can have very high convergence rates, which allow us to use fewer degrees of freedom for a desired level of accuracy. e most common spectral method from the strong form of the equations is known as collocation. In collocation techniques, the partial differential equation must be satisfied at a set of grid, or more precisely, collocation points (see, for instance, [6–10]). Spectral methods also have become increasingly popular for solving fractional differen- tial equations [11–21]. In this paper, we present an accurate numerical solution based on Legendre-Gauss-Lobatto collocation method for Fisher’s type equations. e Fisher equation in the form = + ] (1 − ) (1) was firstly introduced by Fisher in [22] to describe the propa- gation of a mutant gene. Fisher equations have a wide applica- tion in a large number of the chemical kinetics [23], logistic population growth [24], flame propagation [25], population in one-dimensional habitual [26], neutron population in a nuclear reaction [27], neurophysiology [28], branching Brownian motion [23], autocatalytic chemical reactions [29], and nuclear reactor theory [30]. In recent years, many physicists and mathematicians have paid much attention to the Fisher equations due to their importance in mathematical physics. In [31], ¨ Oˇ g¨ un and Kart utilized truncated Painlev` e expansions for presenting some exact solutions of Fisher and generalized Fisher equations. Tan et al. [28] proposed the homotopy analysis method to find analytical solution of Fisher equations. Gunzburger et al. [32] applied the discrete finite element approximation for obtaining a numerical solution of the forced Fisher equation. Dag et al. [33] discussed and applied the B-spline Galerkin method for Fisher’s equation. Bastani and Salkuyeh [34] proposed the compact finite difference approach in combi- nation with third-order Runge-Kutta scheme to solve Fisher’s equation. More recently, Mittal and Jain [35] investigated the cubic B-spline scheme for solving Fisher’s reaction-diffusion problem. However, the fisher equations have been studied in
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 176730 10 pageshttpdxdoiorg1011552013176730
Research ArticleApproximate Solutions of Fisherrsquos Type Equations withVariable Coefficients
A H Bhrawy12 and M A Alghamdi1
1 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia2Department of Mathematics Faculty of Science Beni-Suef University Beni-Suef 62511 Egypt
Correspondence should be addressed to A H Bhrawy alibhrawyyahoocouk
Received 6 September 2013 Accepted 20 September 2013
Academic Editor Dumitru Baleanu
Copyright copy 2013 A H Bhrawy and M A Alghamdi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisherrsquos type problems The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes Theproposed method has the advantage of reducing the problem to a system of ordinary differential equations in time The four-stageA-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time Numerical results show that theLegendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisherrsquos type equations Also theresults demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations
1 Introduction
Spectral methods (see for instance [1ndash5]) are powerful tech-niques that we use to numerically solve linear and nonlinearpartial differential equations either in their strong or weakforms What sets spectral methods apart from others likefinite difference methods or finite element methods is that toget a spectral method we approximate the solutions by highorder orthogonal polynomial expansions The orthogonalpolynomial approximations can have very high convergencerates which allow us to use fewer degrees of freedom for adesired level of accuracyThemost common spectral methodfrom the strong formof the equations is known as collocationIn collocation techniques the partial differential equationmust be satisfied at a set of grid ormore precisely collocationpoints (see for instance [6ndash10]) Spectral methods also havebecome increasingly popular for solving fractional differen-tial equations [11ndash21]
In this paper we present an accurate numerical solutionbased on Legendre-Gauss-Lobatto collocation method forFisherrsquos type equations The Fisher equation in the form
119906119905= 119863119906119909119909
+ ]119906 (1 minus 119906) (1)
was firstly introduced by Fisher in [22] to describe the propa-gation of amutant gene Fisher equations have awide applica-tion in a large number of the chemical kinetics [23] logisticpopulation growth [24] flame propagation [25] populationin one-dimensional habitual [26] neutron population ina nuclear reaction [27] neurophysiology [28] branchingBrownian motion [23] autocatalytic chemical reactions [29]and nuclear reactor theory [30]
In recent years many physicists andmathematicians havepaid much attention to the Fisher equations due to theirimportance in mathematical physics In [31] Ogun and Kartutilized truncated Painleve expansions for presenting someexact solutions of Fisher and generalized Fisher equationsTan et al [28] proposed the homotopy analysis method tofind analytical solution of Fisher equations Gunzburger etal [32] applied the discrete finite element approximation forobtaining a numerical solution of the forced Fisher equationDag et al [33] discussed and applied the B-spline Galerkinmethod for Fisherrsquos equation Bastani and Salkuyeh [34]proposed the compact finite difference approach in combi-nation with third-order Runge-Kutta scheme to solve Fisherrsquosequation More recently Mittal and Jain [35] investigated thecubic B-spline scheme for solving Fisherrsquos reaction-diffusionproblem However the fisher equations have been studied in
2 Abstract and Applied Analysis
many other articles by numerous numerical methods suchas pseudospectral method [36 37] finite difference method[38ndash44] finite element method [45] B-spline algorithm [46]and Galerkin method [47 48]
To increase the numerical solution accuracy spectralcollocation methods based on orthogonal polynomials areoften chosen Doha et al [49] proposed and developed anew numerical algorithm for solving the initial-boundarysystem of nonlinear hyperbolic equations based on spec-tral collocation method a Chebyshev-Gauss-Radau colloca-tion method in combination with the implicit Runge-Kuttascheme are employed to obtain highly accurate approxi-mations to this system of nonlinear hyperbolic equationsIn [50] Bhrawy proposed an efficient Jacobi-Gauss-Lobattocollocation method for approximating the solution of thegeneralized Fitzhugh-Nagumo equation in which the Jacobi-Gauss-Lobatto points are used as collocation nodes forspatial derivatives Moreover the Jacobi spectral collocationmethods are used to solve some problems in mathematicalphysics (see for instance [51ndash53])
Indeed there are no results on Legendre-Gauss-Lobattocollocation method for solving nonlinear Fisher-type equa-tions subject to initial-boundary conditions Therefore theobjective of this work is to present a numerical algorithmfor solving such equation based on Legendre-Gauss-Lobattopseudospectral method The spatial derivatives are approx-imated at these grid points by approximating the deriva-tives of Legendre polynomial that interpolates the solutionsMoreover we set the boundary conditions in the collocationmethod The problem is then reduced to system of first-order ordinary differential equations in time The four-stage A-stable implicit Runge-Kutta scheme is proposed fortreating the this system of equations Finally some illustrativeexamples are implemented to illustrate the efficiency andapplicability of the proposed approach
The rest of this paper is structured as follows In the nextsection some properties of Legendre polynomials whichare required for implementing our algorithm are presentedSection 3 is devoted to the development of Gauss-Lobattocollocation technique for a general form of Fisher-type equa-tions based on the Legendre polynomials and in Section 4 theproposed method is implemented to obtain some numericalresults for three problems of Fisher-type equations withknown exact solutions Finally a brief conclusion is providedin Section 5
2 Legendre Polynomials
The Legendre polynomials 119871119896(119909) (119896 = 0 1 ) satisfy the
following Rodriguesrsquo formula
119871119896(119909) =
(minus1)119896
2119896119896
119863119896((1 minus 119909
2)
119896
) (2)
we recall also that 119871119896(119909) is a polynomial of degree 119896 and
therefore the 119902th derivative of 119871119896(119909) is given by
119871(119902)
119896(119909) =
119896minus119902
sum
119894=0(119896+119894=even)119862119902(119896 119894) 119871
119894(119909) (3)
where
119862119902(119896 119894)
=
2119902minus1
(2119894 + 1) Γ [(119902 + 119896 minus 119894) 2] Γ [(119902 + 119896 + 119894 + 1) 2]
Γ [119902] Γ [(2 minus 119902 + 119896 minus 119894) 2] Γ [(3 minus 119902 + 119896 + 119894) 2]
(4)
The analytical form of Legendre polynomial is
119871119899(119909) =
[1198992]
sum
119894=0
119888(119899)
119896119909119899minus2119896
(5)
where 119888(119899)
119896= (minus1)
119896(2119899 minus 2119896)2
119899(119899 minus 119896)(119899 minus 2119896)119896 and
[
119899
2
] =
119899
2
even119899 minus 1
2
odd(6)
It is also generating from the following relation
119871119896+2
(119909) =
2119896 + 3
119896 + 2
119909119871119896+1
(119909) minus
119896 + 1
119896 + 2
119871119896(119909) (7)
with 1198710(119909) = 1 119871
1(119909) = 119909 and satisfies the orthogonality
condition
(119871119896(119909) 119871
119897(119909))119908= int
1
minus1
119871119896(119909) 119871119897(119909) 119908 (119909) = ℎ
119896120575119897119896 (8)
where 119908(119909) = 1 ℎ119896= 2(2119896 + 1) Let 119878
119873be the space of all
polynomials of degree le 119873 then for any 120601 isin 1198782119873minus1
(0 119871)
int
1
minus1
119908 (119909) 120601 (119909) 119889119909 =
119873
sum
119895=0
120603119873119895
120601 (119909119873119895
) (9)
Let us define the following discrete inner product and norm
(119906 V)119908=
119873
sum
119895=0
119906 (119909119873119895
) V (119909119873119895
) 120603119873119895
(10)
where 119909119873119895
and 120603119873119895
are the nodes and the correspondingweights of the Legendre-Gauss-Lobatto quadrature formulaon the interval (minus1 1) respectively
3 Legendre Spectral Collocation Method
Because of the pseudospectral method is an efficient andaccurate numerical scheme for solving various problems inphysical space including variable coefficient and singularity(see [54 55]) we propose this method based on Legendrepolynomials for approximating the solution of the nonlineargeneralized Burger-Fisher model equation and Fisher modelwith variable coefficient
31 (1+1)-Dimensional Generalized Burger-Fisher EquationIn this subsection we derive a Legendre pseudospectral
Abstract and Applied Analysis 3
algorithm to solve numerically the generalized Burger-Fisherproblem
119906119905+ ]119906120575119906
119909minus 119906119909119909
minus 120574119906 (1 minus 119906120575) = 0 (119909 119905) isin 119863 times [0 119879]
(11)
where 119863 = 119909 minus1 le 119909 le 1 Subject to
119906 (119909 119905) = 119892 (119905) 119909 = minus1 1 (12)
119906 (119909 0) = 119891 (119909) 119909 isin 119863 (13)
In the following we shall derive an efficient algorithm forthe numerical solution of (11)ndash(13) Let the approximationof 119906(119909 119905) be given in terms of the Legendre polynomialsexpansion
119906 (119909 119905) =
119873
sum
119895=0
119886119895(119905) 119871119895(119909) a = (119886
0 1198861 119886
119873)119879
(14)
Making use of relations (8) and (10) gives
119906 (119909 119905) =
119873
sum
119895=0
(
1
ℎ119895
119873
sum
119894=0
119871119895(119909119894) 119871119895(119909) 120603119873119894
119906 (119909119894 119905)) (15)
or equivalently
119906 (119909 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) 119871119895(119909) 120603119873119894
)119906 (119909119894 119905) (16)
The Gauss-Lobatto points were introduced by wayof (9) We then saw that the polynomial approximation119906(119909 119905) can be characterized by (119873 + 1) nodal values 119906(119909
119894 119905)
The approximation of the spatial partial derivatives of first-order for 119906(119909 119905) can be computed at the Legendre Gauss-Lobatto interpolation nodes as
119906119909(119909119899 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
1015840
120603119873119894
)119906 (119909119894 119905)
=
119873
sum
119894=0
119860119899119894119906 (119909119894 119905)
=
119873
sum
119894=0
119860119899119894119906119894(119905) 119899 = 0 1 119873
(17)
where
119860119899119894=
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
1015840
120603119873119894
119906119894(119905) = 119906 (119909
119894 119905)
(18)
Subsequently the second-order spatial partial derivativesof 119906(119909 119905) may be written at the same collocation nodes as
119906119909119909
(119909119899 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
10158401015840
120603119873119894
)119906 (119909119894 119905)
=
119873
sum
119894=0
119861119899119894119906 (119909119894 119905)
=
119873
sum
119894=0
119861119899119894119906119894
(19)
where
119861119899119894=
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
10158401015840
120603119873119894
(20)
In collocationmethods one specifically seeks the approx-imate solution such that the problem (11) is satisfied exactlyat the Legendre Gauss-Lobatto set of interpolation points 119909
119899
119899 = 1 119873 minus 1 The approximation is exact at the 119873 minus
1 collocation pointsTherefore (11) after using relations (17)ndash(20) can be written as
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873
sum
119894=0
119860119899119894119906119894(119905)
minus
119873
sum
119894=0
119861119899119894119906119894(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
(21)
where 119906119899(119905) = 119906(119909
119899 119905) and 119906
sdot
119899(119905) = 120597119906
sdot
119899(119905)120597119905
Now the two values 1199060(119905) and 119906
119873(119905) can be determined
from the boundary conditions (12) then (21) can be reformu-lated as
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905)
minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + ]119906120575
119899(119905) 119889119899(119905) minus
119889119899(119905)
minus 120574119906119899(119905) (1 minus 119906
120575
119899(119905)) = 0 119899 = 1 119873 minus 1
(22)
where
119889119899(119905) = 119860
11989901199060(119905) + 119860
119899119873119906119873(119905)
119889119899(119905) = 119861
11989901199060(119905) + 119861
119899119873119906119873(119905)
(23)
Approximation (22) automatically satisfies the boundaryconditions (12) but we need an initial condition for eachof the 119906
119899(119905) to integrate (22) in time The initial condi-
tion is usually taken to be the interpolant of the initial
4 Abstract and Applied Analysis
function 119891(119909) that is 119906119899(0) = 119891(119909
119899) Therefore the approx-
imation of (11)ndash(13) is reduced to the solution of system ofordinary differential equations in time Consider
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
+ ]119906120575119899(119905) 119889119899(119905) minus
119889119899(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
119906119899(0) = 119891 (119909
119899)
(24)
Let us denote
119880sdot(119905) = [119906
sdot
1(119905) 119906sdot
2(119905) 119906
sdot
119873minus1(119905)]119879
119880 (0) = [1199061(0) 1199062(0) 119906
119873minus1(0)]119879
119891 = [119891(1199091) 119891(119909
2) 119891(119909
119873minus1)]119879
119865 (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865
2(119905 119906 (119905)) 119865
119873minus1(119905 119906 (119905))]
119879
119865119899(119905 119906 (119905)) = minus ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) +
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
minus ]119906120575119899(119905) 119889119899(119905) +
119889119899(119905) + 120574119906
119899(119905) (1 minus 119906
120575
119899(119905))
119899 = 1 119873 minus 1
(25)
Then (24) can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(26)
This system of ordinary differential equations can be solvedby using four-stage A-stable implicit Runge-Kutta scheme
32 (1+1)-Dimensional Fisher Equation with Variable Coef-ficient In this subsection we extend the application of theLegendre pseudospectral method to solve numerically theFisher equation with variable coefficient
119906119905minus 119887 (119905) 119906
119909119909minus 119888119906 (1 minus 119906) = 0 (119909 119905) isin 119863 times [0 119879] (27)
subject to the initial-boundary conditions
119906 (119909 0) = 119891 (119909) 119909 isin 119863
119906 (119909 119905) = 119892 (119905)
(28)
Proceeding as in the previous subsection we can obtain 119906119909119909
in the same form as (19) and then (27) can be collocated inthe Legendre Gauss-Lobatto points as
119906sdot
119899(119905) minus 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) minus 119887 (119905)
119889119899(119905) minus 119888119906
119899(119905) (1 minus 119906
119899(119905))
= 0 119899 = 1 119873 minus 1
(29)
Table 1 Absolute errors for Example 1
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
556 times 10minus11
minus1
02
893 times 10minus11
minus05 146 times 10minus8
minus05 823 times 10minus9
0 195 times 10minus8 0 141 times 10
minus8
05 155 times 10minus8 05 117 times 10
minus8
1 556 times 10minus11 1 893 times 10
minus11
which can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(30)
where
119865119899(119905 119906 (119905))
= 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + 119887 (119905)
119889119899(119905) + 119888119906
119899(119905) (1 minus 119906
119899(119905))
119899 = 1 119873 minus 1
(31)
4 Numerical Examples
In this section three nonlinear time-dependent Fisher-typeequations on finite interval are implemented to demonstratethe accuracy and capability of the proposed algorithm and allof them were performed on the computer using a programwritten in Mathematica 80 The absolute errors in thegiven tables are 119864(119909 119905) = |119906(119909 119905) minus (119909 119905) where 119906(119909 119905)
and (119909 119905) are the exact and numerical solution at selectedpoints (119909 119905)
Example 1 Consider the nonlinear time-dependent one-dimensional Fisher-type equations
119906119905= 119906119909119909
+ 119906 (1 minus 119906) (119906 minus 120574) (119909 119905) isin 119863 times [0 119879] (32)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 minus
1 + 120574
radic2
119905)]
119906 (minus1 119905) =
1 + 120574
2
+
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 +
1 + 120574
radic2
119905)]
119906 (119909 0) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909)] 119909 isin 119863
(33)
The exact solution is
119906 (119909 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909 minus
1 + 120574
radic2
119905)] (34)
In Table 1 we introduce the absolute errors between theapproximate and exact solutions for problem (32) using theproposed method for different values of 119909 and 119905 with 120574 =
10minus2 and 119873 = 20
Abstract and Applied Analysis 5
07
06
05
04
minus10
minus05
00
05
10
10
05
00
u
t
x
(a)
00
05
10
t
x
E
4
times10minus8
3
2
1
0
minus10
minus05
00
05
10
(b)
Figure 1 The result of the L-GL-C method at 120574 = 10minus2 and 119873 = 20 (a) The approximate solution (b) The absolute error
100500
05
minus10
04
minus05
06
07
08
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 2 The curves of approximate solutions and the exactsolutions of problem (32) at 119905 = 00 119905 = 05 and 119905 = 09 with 120574 =
10minus2 and 119873 = 20
In case of 120574 = 10minus2 and 119873 = 20 the approximate solu-
tion and absolute errors of problem (32) are displayed inFigures 1(a) and 1(b) respectively In Figure 2 we plotted thecurves of approximate solutions and exact solutions of pro-blem (32) for different values of 119905 (0005 and 09) with 120574 =
10minus2 and119873 = 20 It is clear from this figure that approximate
solutions and exact solutions completely coincide for thechosen values of 119905
Example 2 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equations
119906119905= 119906119909119909
minus ]119906120575119906119909+ 120574119906 (1 minus 119906
120575) (119909 119905) isin 119863 times [0 119879]
(35)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905)
= (
1
2
minus
1
2
tanh [
]120575
2 (120575 + 1)
times (1 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (minus1 119905)
= (
1
2
+
1
2
tanh [ ]120575
2 (120575 + 1)
times (1 + (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (119909 0) = (
1
2
minus
1
2
tanh [ ]120575
2 (120575 + 1)
119909])
1120575
119909 isin 119863
(36)
6 Abstract and Applied Analysis
0502
minus10
minus05
00
05
10 00
05
10
x
u
t
0500
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
8
times10minus10
6
4
2
0
(b)
Figure 3 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 1 and 119873 = 20 (a) The approximate solution (b)The absolute error
05005
05010
05015
05020
05025
05030
100500minus10 minus05
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 4 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 1 and119873 = 20
The exact solution of (35) is
119906 (119909 119905)
= (
1
2
minus
1
2
tanh( ]120575
2 (120575 + 1)
times(119909 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)))
1120575
(37)
The absolute errors for problem (35) are listed in Table 2using the L-GL-C method with ] = 120574 = 10
minus2 119873 = 20 andvarious choices of 120575
To illustrate the effectiveness of the Legendre pseudospec-tralmethod for problem (35) we displayed in Figures 3(a) and3(b) the approximate solution and the absolute error with ] =
120574 = 10minus2 120575 = 1 and 119873 = 20 The graph of curves of exact
and approximate solutions with different values of 119905 (0005 and 09) is given in Figure 4 Moreover the approximatesolution and the absolute error with ] = 120574 = 10
minus2 120575 = 2 and119873 = 20 are displayed in Figures 5(a) and 5(b) respectivelyThe curves of exact and approximate solutions of problem(35) with 120575 = 2 are displayed in Figure 6 with values ofparameters listed in its caption
Example 3 Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient
119906119905= minus
119886
61205832coth(119886
6
119905 + 119888) 119906119909119909
+ 119886119906 (1 minus 119906)
(119909 119905) isin 119863 times [0 119879]
(38)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583
2
+
5119886
12
119905)
+
1
2
tanh(120583
2
+
5119886
12
119905) +
1
2
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
many other articles by numerous numerical methods suchas pseudospectral method [36 37] finite difference method[38ndash44] finite element method [45] B-spline algorithm [46]and Galerkin method [47 48]
To increase the numerical solution accuracy spectralcollocation methods based on orthogonal polynomials areoften chosen Doha et al [49] proposed and developed anew numerical algorithm for solving the initial-boundarysystem of nonlinear hyperbolic equations based on spec-tral collocation method a Chebyshev-Gauss-Radau colloca-tion method in combination with the implicit Runge-Kuttascheme are employed to obtain highly accurate approxi-mations to this system of nonlinear hyperbolic equationsIn [50] Bhrawy proposed an efficient Jacobi-Gauss-Lobattocollocation method for approximating the solution of thegeneralized Fitzhugh-Nagumo equation in which the Jacobi-Gauss-Lobatto points are used as collocation nodes forspatial derivatives Moreover the Jacobi spectral collocationmethods are used to solve some problems in mathematicalphysics (see for instance [51ndash53])
Indeed there are no results on Legendre-Gauss-Lobattocollocation method for solving nonlinear Fisher-type equa-tions subject to initial-boundary conditions Therefore theobjective of this work is to present a numerical algorithmfor solving such equation based on Legendre-Gauss-Lobattopseudospectral method The spatial derivatives are approx-imated at these grid points by approximating the deriva-tives of Legendre polynomial that interpolates the solutionsMoreover we set the boundary conditions in the collocationmethod The problem is then reduced to system of first-order ordinary differential equations in time The four-stage A-stable implicit Runge-Kutta scheme is proposed fortreating the this system of equations Finally some illustrativeexamples are implemented to illustrate the efficiency andapplicability of the proposed approach
The rest of this paper is structured as follows In the nextsection some properties of Legendre polynomials whichare required for implementing our algorithm are presentedSection 3 is devoted to the development of Gauss-Lobattocollocation technique for a general form of Fisher-type equa-tions based on the Legendre polynomials and in Section 4 theproposed method is implemented to obtain some numericalresults for three problems of Fisher-type equations withknown exact solutions Finally a brief conclusion is providedin Section 5
2 Legendre Polynomials
The Legendre polynomials 119871119896(119909) (119896 = 0 1 ) satisfy the
following Rodriguesrsquo formula
119871119896(119909) =
(minus1)119896
2119896119896
119863119896((1 minus 119909
2)
119896
) (2)
we recall also that 119871119896(119909) is a polynomial of degree 119896 and
therefore the 119902th derivative of 119871119896(119909) is given by
119871(119902)
119896(119909) =
119896minus119902
sum
119894=0(119896+119894=even)119862119902(119896 119894) 119871
119894(119909) (3)
where
119862119902(119896 119894)
=
2119902minus1
(2119894 + 1) Γ [(119902 + 119896 minus 119894) 2] Γ [(119902 + 119896 + 119894 + 1) 2]
Γ [119902] Γ [(2 minus 119902 + 119896 minus 119894) 2] Γ [(3 minus 119902 + 119896 + 119894) 2]
(4)
The analytical form of Legendre polynomial is
119871119899(119909) =
[1198992]
sum
119894=0
119888(119899)
119896119909119899minus2119896
(5)
where 119888(119899)
119896= (minus1)
119896(2119899 minus 2119896)2
119899(119899 minus 119896)(119899 minus 2119896)119896 and
[
119899
2
] =
119899
2
even119899 minus 1
2
odd(6)
It is also generating from the following relation
119871119896+2
(119909) =
2119896 + 3
119896 + 2
119909119871119896+1
(119909) minus
119896 + 1
119896 + 2
119871119896(119909) (7)
with 1198710(119909) = 1 119871
1(119909) = 119909 and satisfies the orthogonality
condition
(119871119896(119909) 119871
119897(119909))119908= int
1
minus1
119871119896(119909) 119871119897(119909) 119908 (119909) = ℎ
119896120575119897119896 (8)
where 119908(119909) = 1 ℎ119896= 2(2119896 + 1) Let 119878
119873be the space of all
polynomials of degree le 119873 then for any 120601 isin 1198782119873minus1
(0 119871)
int
1
minus1
119908 (119909) 120601 (119909) 119889119909 =
119873
sum
119895=0
120603119873119895
120601 (119909119873119895
) (9)
Let us define the following discrete inner product and norm
(119906 V)119908=
119873
sum
119895=0
119906 (119909119873119895
) V (119909119873119895
) 120603119873119895
(10)
where 119909119873119895
and 120603119873119895
are the nodes and the correspondingweights of the Legendre-Gauss-Lobatto quadrature formulaon the interval (minus1 1) respectively
3 Legendre Spectral Collocation Method
Because of the pseudospectral method is an efficient andaccurate numerical scheme for solving various problems inphysical space including variable coefficient and singularity(see [54 55]) we propose this method based on Legendrepolynomials for approximating the solution of the nonlineargeneralized Burger-Fisher model equation and Fisher modelwith variable coefficient
31 (1+1)-Dimensional Generalized Burger-Fisher EquationIn this subsection we derive a Legendre pseudospectral
Abstract and Applied Analysis 3
algorithm to solve numerically the generalized Burger-Fisherproblem
119906119905+ ]119906120575119906
119909minus 119906119909119909
minus 120574119906 (1 minus 119906120575) = 0 (119909 119905) isin 119863 times [0 119879]
(11)
where 119863 = 119909 minus1 le 119909 le 1 Subject to
119906 (119909 119905) = 119892 (119905) 119909 = minus1 1 (12)
119906 (119909 0) = 119891 (119909) 119909 isin 119863 (13)
In the following we shall derive an efficient algorithm forthe numerical solution of (11)ndash(13) Let the approximationof 119906(119909 119905) be given in terms of the Legendre polynomialsexpansion
119906 (119909 119905) =
119873
sum
119895=0
119886119895(119905) 119871119895(119909) a = (119886
0 1198861 119886
119873)119879
(14)
Making use of relations (8) and (10) gives
119906 (119909 119905) =
119873
sum
119895=0
(
1
ℎ119895
119873
sum
119894=0
119871119895(119909119894) 119871119895(119909) 120603119873119894
119906 (119909119894 119905)) (15)
or equivalently
119906 (119909 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) 119871119895(119909) 120603119873119894
)119906 (119909119894 119905) (16)
The Gauss-Lobatto points were introduced by wayof (9) We then saw that the polynomial approximation119906(119909 119905) can be characterized by (119873 + 1) nodal values 119906(119909
119894 119905)
The approximation of the spatial partial derivatives of first-order for 119906(119909 119905) can be computed at the Legendre Gauss-Lobatto interpolation nodes as
119906119909(119909119899 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
1015840
120603119873119894
)119906 (119909119894 119905)
=
119873
sum
119894=0
119860119899119894119906 (119909119894 119905)
=
119873
sum
119894=0
119860119899119894119906119894(119905) 119899 = 0 1 119873
(17)
where
119860119899119894=
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
1015840
120603119873119894
119906119894(119905) = 119906 (119909
119894 119905)
(18)
Subsequently the second-order spatial partial derivativesof 119906(119909 119905) may be written at the same collocation nodes as
119906119909119909
(119909119899 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
10158401015840
120603119873119894
)119906 (119909119894 119905)
=
119873
sum
119894=0
119861119899119894119906 (119909119894 119905)
=
119873
sum
119894=0
119861119899119894119906119894
(19)
where
119861119899119894=
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
10158401015840
120603119873119894
(20)
In collocationmethods one specifically seeks the approx-imate solution such that the problem (11) is satisfied exactlyat the Legendre Gauss-Lobatto set of interpolation points 119909
119899
119899 = 1 119873 minus 1 The approximation is exact at the 119873 minus
1 collocation pointsTherefore (11) after using relations (17)ndash(20) can be written as
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873
sum
119894=0
119860119899119894119906119894(119905)
minus
119873
sum
119894=0
119861119899119894119906119894(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
(21)
where 119906119899(119905) = 119906(119909
119899 119905) and 119906
sdot
119899(119905) = 120597119906
sdot
119899(119905)120597119905
Now the two values 1199060(119905) and 119906
119873(119905) can be determined
from the boundary conditions (12) then (21) can be reformu-lated as
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905)
minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + ]119906120575
119899(119905) 119889119899(119905) minus
119889119899(119905)
minus 120574119906119899(119905) (1 minus 119906
120575
119899(119905)) = 0 119899 = 1 119873 minus 1
(22)
where
119889119899(119905) = 119860
11989901199060(119905) + 119860
119899119873119906119873(119905)
119889119899(119905) = 119861
11989901199060(119905) + 119861
119899119873119906119873(119905)
(23)
Approximation (22) automatically satisfies the boundaryconditions (12) but we need an initial condition for eachof the 119906
119899(119905) to integrate (22) in time The initial condi-
tion is usually taken to be the interpolant of the initial
4 Abstract and Applied Analysis
function 119891(119909) that is 119906119899(0) = 119891(119909
119899) Therefore the approx-
imation of (11)ndash(13) is reduced to the solution of system ofordinary differential equations in time Consider
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
+ ]119906120575119899(119905) 119889119899(119905) minus
119889119899(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
119906119899(0) = 119891 (119909
119899)
(24)
Let us denote
119880sdot(119905) = [119906
sdot
1(119905) 119906sdot
2(119905) 119906
sdot
119873minus1(119905)]119879
119880 (0) = [1199061(0) 1199062(0) 119906
119873minus1(0)]119879
119891 = [119891(1199091) 119891(119909
2) 119891(119909
119873minus1)]119879
119865 (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865
2(119905 119906 (119905)) 119865
119873minus1(119905 119906 (119905))]
119879
119865119899(119905 119906 (119905)) = minus ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) +
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
minus ]119906120575119899(119905) 119889119899(119905) +
119889119899(119905) + 120574119906
119899(119905) (1 minus 119906
120575
119899(119905))
119899 = 1 119873 minus 1
(25)
Then (24) can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(26)
This system of ordinary differential equations can be solvedby using four-stage A-stable implicit Runge-Kutta scheme
32 (1+1)-Dimensional Fisher Equation with Variable Coef-ficient In this subsection we extend the application of theLegendre pseudospectral method to solve numerically theFisher equation with variable coefficient
119906119905minus 119887 (119905) 119906
119909119909minus 119888119906 (1 minus 119906) = 0 (119909 119905) isin 119863 times [0 119879] (27)
subject to the initial-boundary conditions
119906 (119909 0) = 119891 (119909) 119909 isin 119863
119906 (119909 119905) = 119892 (119905)
(28)
Proceeding as in the previous subsection we can obtain 119906119909119909
in the same form as (19) and then (27) can be collocated inthe Legendre Gauss-Lobatto points as
119906sdot
119899(119905) minus 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) minus 119887 (119905)
119889119899(119905) minus 119888119906
119899(119905) (1 minus 119906
119899(119905))
= 0 119899 = 1 119873 minus 1
(29)
Table 1 Absolute errors for Example 1
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
556 times 10minus11
minus1
02
893 times 10minus11
minus05 146 times 10minus8
minus05 823 times 10minus9
0 195 times 10minus8 0 141 times 10
minus8
05 155 times 10minus8 05 117 times 10
minus8
1 556 times 10minus11 1 893 times 10
minus11
which can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(30)
where
119865119899(119905 119906 (119905))
= 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + 119887 (119905)
119889119899(119905) + 119888119906
119899(119905) (1 minus 119906
119899(119905))
119899 = 1 119873 minus 1
(31)
4 Numerical Examples
In this section three nonlinear time-dependent Fisher-typeequations on finite interval are implemented to demonstratethe accuracy and capability of the proposed algorithm and allof them were performed on the computer using a programwritten in Mathematica 80 The absolute errors in thegiven tables are 119864(119909 119905) = |119906(119909 119905) minus (119909 119905) where 119906(119909 119905)
and (119909 119905) are the exact and numerical solution at selectedpoints (119909 119905)
Example 1 Consider the nonlinear time-dependent one-dimensional Fisher-type equations
119906119905= 119906119909119909
+ 119906 (1 minus 119906) (119906 minus 120574) (119909 119905) isin 119863 times [0 119879] (32)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 minus
1 + 120574
radic2
119905)]
119906 (minus1 119905) =
1 + 120574
2
+
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 +
1 + 120574
radic2
119905)]
119906 (119909 0) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909)] 119909 isin 119863
(33)
The exact solution is
119906 (119909 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909 minus
1 + 120574
radic2
119905)] (34)
In Table 1 we introduce the absolute errors between theapproximate and exact solutions for problem (32) using theproposed method for different values of 119909 and 119905 with 120574 =
10minus2 and 119873 = 20
Abstract and Applied Analysis 5
07
06
05
04
minus10
minus05
00
05
10
10
05
00
u
t
x
(a)
00
05
10
t
x
E
4
times10minus8
3
2
1
0
minus10
minus05
00
05
10
(b)
Figure 1 The result of the L-GL-C method at 120574 = 10minus2 and 119873 = 20 (a) The approximate solution (b) The absolute error
100500
05
minus10
04
minus05
06
07
08
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 2 The curves of approximate solutions and the exactsolutions of problem (32) at 119905 = 00 119905 = 05 and 119905 = 09 with 120574 =
10minus2 and 119873 = 20
In case of 120574 = 10minus2 and 119873 = 20 the approximate solu-
tion and absolute errors of problem (32) are displayed inFigures 1(a) and 1(b) respectively In Figure 2 we plotted thecurves of approximate solutions and exact solutions of pro-blem (32) for different values of 119905 (0005 and 09) with 120574 =
10minus2 and119873 = 20 It is clear from this figure that approximate
solutions and exact solutions completely coincide for thechosen values of 119905
Example 2 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equations
119906119905= 119906119909119909
minus ]119906120575119906119909+ 120574119906 (1 minus 119906
120575) (119909 119905) isin 119863 times [0 119879]
(35)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905)
= (
1
2
minus
1
2
tanh [
]120575
2 (120575 + 1)
times (1 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (minus1 119905)
= (
1
2
+
1
2
tanh [ ]120575
2 (120575 + 1)
times (1 + (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (119909 0) = (
1
2
minus
1
2
tanh [ ]120575
2 (120575 + 1)
119909])
1120575
119909 isin 119863
(36)
6 Abstract and Applied Analysis
0502
minus10
minus05
00
05
10 00
05
10
x
u
t
0500
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
8
times10minus10
6
4
2
0
(b)
Figure 3 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 1 and 119873 = 20 (a) The approximate solution (b)The absolute error
05005
05010
05015
05020
05025
05030
100500minus10 minus05
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 4 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 1 and119873 = 20
The exact solution of (35) is
119906 (119909 119905)
= (
1
2
minus
1
2
tanh( ]120575
2 (120575 + 1)
times(119909 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)))
1120575
(37)
The absolute errors for problem (35) are listed in Table 2using the L-GL-C method with ] = 120574 = 10
minus2 119873 = 20 andvarious choices of 120575
To illustrate the effectiveness of the Legendre pseudospec-tralmethod for problem (35) we displayed in Figures 3(a) and3(b) the approximate solution and the absolute error with ] =
120574 = 10minus2 120575 = 1 and 119873 = 20 The graph of curves of exact
and approximate solutions with different values of 119905 (0005 and 09) is given in Figure 4 Moreover the approximatesolution and the absolute error with ] = 120574 = 10
minus2 120575 = 2 and119873 = 20 are displayed in Figures 5(a) and 5(b) respectivelyThe curves of exact and approximate solutions of problem(35) with 120575 = 2 are displayed in Figure 6 with values ofparameters listed in its caption
Example 3 Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient
119906119905= minus
119886
61205832coth(119886
6
119905 + 119888) 119906119909119909
+ 119886119906 (1 minus 119906)
(119909 119905) isin 119863 times [0 119879]
(38)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583
2
+
5119886
12
119905)
+
1
2
tanh(120583
2
+
5119886
12
119905) +
1
2
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
algorithm to solve numerically the generalized Burger-Fisherproblem
119906119905+ ]119906120575119906
119909minus 119906119909119909
minus 120574119906 (1 minus 119906120575) = 0 (119909 119905) isin 119863 times [0 119879]
(11)
where 119863 = 119909 minus1 le 119909 le 1 Subject to
119906 (119909 119905) = 119892 (119905) 119909 = minus1 1 (12)
119906 (119909 0) = 119891 (119909) 119909 isin 119863 (13)
In the following we shall derive an efficient algorithm forthe numerical solution of (11)ndash(13) Let the approximationof 119906(119909 119905) be given in terms of the Legendre polynomialsexpansion
119906 (119909 119905) =
119873
sum
119895=0
119886119895(119905) 119871119895(119909) a = (119886
0 1198861 119886
119873)119879
(14)
Making use of relations (8) and (10) gives
119906 (119909 119905) =
119873
sum
119895=0
(
1
ℎ119895
119873
sum
119894=0
119871119895(119909119894) 119871119895(119909) 120603119873119894
119906 (119909119894 119905)) (15)
or equivalently
119906 (119909 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) 119871119895(119909) 120603119873119894
)119906 (119909119894 119905) (16)
The Gauss-Lobatto points were introduced by wayof (9) We then saw that the polynomial approximation119906(119909 119905) can be characterized by (119873 + 1) nodal values 119906(119909
119894 119905)
The approximation of the spatial partial derivatives of first-order for 119906(119909 119905) can be computed at the Legendre Gauss-Lobatto interpolation nodes as
119906119909(119909119899 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
1015840
120603119873119894
)119906 (119909119894 119905)
=
119873
sum
119894=0
119860119899119894119906 (119909119894 119905)
=
119873
sum
119894=0
119860119899119894119906119894(119905) 119899 = 0 1 119873
(17)
where
119860119899119894=
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
1015840
120603119873119894
119906119894(119905) = 119906 (119909
119894 119905)
(18)
Subsequently the second-order spatial partial derivativesof 119906(119909 119905) may be written at the same collocation nodes as
119906119909119909
(119909119899 119905) =
119873
sum
119894=0
(
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
10158401015840
120603119873119894
)119906 (119909119894 119905)
=
119873
sum
119894=0
119861119899119894119906 (119909119894 119905)
=
119873
sum
119894=0
119861119899119894119906119894
(19)
where
119861119899119894=
119873
sum
119895=0
1
ℎ119895
119871119895(119909119894) (119871119895(119909119899))
10158401015840
120603119873119894
(20)
In collocationmethods one specifically seeks the approx-imate solution such that the problem (11) is satisfied exactlyat the Legendre Gauss-Lobatto set of interpolation points 119909
119899
119899 = 1 119873 minus 1 The approximation is exact at the 119873 minus
1 collocation pointsTherefore (11) after using relations (17)ndash(20) can be written as
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873
sum
119894=0
119860119899119894119906119894(119905)
minus
119873
sum
119894=0
119861119899119894119906119894(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
(21)
where 119906119899(119905) = 119906(119909
119899 119905) and 119906
sdot
119899(119905) = 120597119906
sdot
119899(119905)120597119905
Now the two values 1199060(119905) and 119906
119873(119905) can be determined
from the boundary conditions (12) then (21) can be reformu-lated as
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905)
minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + ]119906120575
119899(119905) 119889119899(119905) minus
119889119899(119905)
minus 120574119906119899(119905) (1 minus 119906
120575
119899(119905)) = 0 119899 = 1 119873 minus 1
(22)
where
119889119899(119905) = 119860
11989901199060(119905) + 119860
119899119873119906119873(119905)
119889119899(119905) = 119861
11989901199060(119905) + 119861
119899119873119906119873(119905)
(23)
Approximation (22) automatically satisfies the boundaryconditions (12) but we need an initial condition for eachof the 119906
119899(119905) to integrate (22) in time The initial condi-
tion is usually taken to be the interpolant of the initial
4 Abstract and Applied Analysis
function 119891(119909) that is 119906119899(0) = 119891(119909
119899) Therefore the approx-
imation of (11)ndash(13) is reduced to the solution of system ofordinary differential equations in time Consider
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
+ ]119906120575119899(119905) 119889119899(119905) minus
119889119899(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
119906119899(0) = 119891 (119909
119899)
(24)
Let us denote
119880sdot(119905) = [119906
sdot
1(119905) 119906sdot
2(119905) 119906
sdot
119873minus1(119905)]119879
119880 (0) = [1199061(0) 1199062(0) 119906
119873minus1(0)]119879
119891 = [119891(1199091) 119891(119909
2) 119891(119909
119873minus1)]119879
119865 (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865
2(119905 119906 (119905)) 119865
119873minus1(119905 119906 (119905))]
119879
119865119899(119905 119906 (119905)) = minus ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) +
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
minus ]119906120575119899(119905) 119889119899(119905) +
119889119899(119905) + 120574119906
119899(119905) (1 minus 119906
120575
119899(119905))
119899 = 1 119873 minus 1
(25)
Then (24) can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(26)
This system of ordinary differential equations can be solvedby using four-stage A-stable implicit Runge-Kutta scheme
32 (1+1)-Dimensional Fisher Equation with Variable Coef-ficient In this subsection we extend the application of theLegendre pseudospectral method to solve numerically theFisher equation with variable coefficient
119906119905minus 119887 (119905) 119906
119909119909minus 119888119906 (1 minus 119906) = 0 (119909 119905) isin 119863 times [0 119879] (27)
subject to the initial-boundary conditions
119906 (119909 0) = 119891 (119909) 119909 isin 119863
119906 (119909 119905) = 119892 (119905)
(28)
Proceeding as in the previous subsection we can obtain 119906119909119909
in the same form as (19) and then (27) can be collocated inthe Legendre Gauss-Lobatto points as
119906sdot
119899(119905) minus 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) minus 119887 (119905)
119889119899(119905) minus 119888119906
119899(119905) (1 minus 119906
119899(119905))
= 0 119899 = 1 119873 minus 1
(29)
Table 1 Absolute errors for Example 1
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
556 times 10minus11
minus1
02
893 times 10minus11
minus05 146 times 10minus8
minus05 823 times 10minus9
0 195 times 10minus8 0 141 times 10
minus8
05 155 times 10minus8 05 117 times 10
minus8
1 556 times 10minus11 1 893 times 10
minus11
which can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(30)
where
119865119899(119905 119906 (119905))
= 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + 119887 (119905)
119889119899(119905) + 119888119906
119899(119905) (1 minus 119906
119899(119905))
119899 = 1 119873 minus 1
(31)
4 Numerical Examples
In this section three nonlinear time-dependent Fisher-typeequations on finite interval are implemented to demonstratethe accuracy and capability of the proposed algorithm and allof them were performed on the computer using a programwritten in Mathematica 80 The absolute errors in thegiven tables are 119864(119909 119905) = |119906(119909 119905) minus (119909 119905) where 119906(119909 119905)
and (119909 119905) are the exact and numerical solution at selectedpoints (119909 119905)
Example 1 Consider the nonlinear time-dependent one-dimensional Fisher-type equations
119906119905= 119906119909119909
+ 119906 (1 minus 119906) (119906 minus 120574) (119909 119905) isin 119863 times [0 119879] (32)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 minus
1 + 120574
radic2
119905)]
119906 (minus1 119905) =
1 + 120574
2
+
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 +
1 + 120574
radic2
119905)]
119906 (119909 0) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909)] 119909 isin 119863
(33)
The exact solution is
119906 (119909 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909 minus
1 + 120574
radic2
119905)] (34)
In Table 1 we introduce the absolute errors between theapproximate and exact solutions for problem (32) using theproposed method for different values of 119909 and 119905 with 120574 =
10minus2 and 119873 = 20
Abstract and Applied Analysis 5
07
06
05
04
minus10
minus05
00
05
10
10
05
00
u
t
x
(a)
00
05
10
t
x
E
4
times10minus8
3
2
1
0
minus10
minus05
00
05
10
(b)
Figure 1 The result of the L-GL-C method at 120574 = 10minus2 and 119873 = 20 (a) The approximate solution (b) The absolute error
100500
05
minus10
04
minus05
06
07
08
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 2 The curves of approximate solutions and the exactsolutions of problem (32) at 119905 = 00 119905 = 05 and 119905 = 09 with 120574 =
10minus2 and 119873 = 20
In case of 120574 = 10minus2 and 119873 = 20 the approximate solu-
tion and absolute errors of problem (32) are displayed inFigures 1(a) and 1(b) respectively In Figure 2 we plotted thecurves of approximate solutions and exact solutions of pro-blem (32) for different values of 119905 (0005 and 09) with 120574 =
10minus2 and119873 = 20 It is clear from this figure that approximate
solutions and exact solutions completely coincide for thechosen values of 119905
Example 2 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equations
119906119905= 119906119909119909
minus ]119906120575119906119909+ 120574119906 (1 minus 119906
120575) (119909 119905) isin 119863 times [0 119879]
(35)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905)
= (
1
2
minus
1
2
tanh [
]120575
2 (120575 + 1)
times (1 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (minus1 119905)
= (
1
2
+
1
2
tanh [ ]120575
2 (120575 + 1)
times (1 + (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (119909 0) = (
1
2
minus
1
2
tanh [ ]120575
2 (120575 + 1)
119909])
1120575
119909 isin 119863
(36)
6 Abstract and Applied Analysis
0502
minus10
minus05
00
05
10 00
05
10
x
u
t
0500
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
8
times10minus10
6
4
2
0
(b)
Figure 3 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 1 and 119873 = 20 (a) The approximate solution (b)The absolute error
05005
05010
05015
05020
05025
05030
100500minus10 minus05
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 4 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 1 and119873 = 20
The exact solution of (35) is
119906 (119909 119905)
= (
1
2
minus
1
2
tanh( ]120575
2 (120575 + 1)
times(119909 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)))
1120575
(37)
The absolute errors for problem (35) are listed in Table 2using the L-GL-C method with ] = 120574 = 10
minus2 119873 = 20 andvarious choices of 120575
To illustrate the effectiveness of the Legendre pseudospec-tralmethod for problem (35) we displayed in Figures 3(a) and3(b) the approximate solution and the absolute error with ] =
120574 = 10minus2 120575 = 1 and 119873 = 20 The graph of curves of exact
and approximate solutions with different values of 119905 (0005 and 09) is given in Figure 4 Moreover the approximatesolution and the absolute error with ] = 120574 = 10
minus2 120575 = 2 and119873 = 20 are displayed in Figures 5(a) and 5(b) respectivelyThe curves of exact and approximate solutions of problem(35) with 120575 = 2 are displayed in Figure 6 with values ofparameters listed in its caption
Example 3 Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient
119906119905= minus
119886
61205832coth(119886
6
119905 + 119888) 119906119909119909
+ 119886119906 (1 minus 119906)
(119909 119905) isin 119863 times [0 119879]
(38)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583
2
+
5119886
12
119905)
+
1
2
tanh(120583
2
+
5119886
12
119905) +
1
2
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
function 119891(119909) that is 119906119899(0) = 119891(119909
119899) Therefore the approx-
imation of (11)ndash(13) is reduced to the solution of system ofordinary differential equations in time Consider
119906sdot
119899(119905) + ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) minus
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
+ ]119906120575119899(119905) 119889119899(119905) minus
119889119899(119905) minus 120574119906
119899(119905) (1 minus 119906
120575
119899(119905)) = 0
119899 = 1 119873 minus 1
119906119899(0) = 119891 (119909
119899)
(24)
Let us denote
119880sdot(119905) = [119906
sdot
1(119905) 119906sdot
2(119905) 119906
sdot
119873minus1(119905)]119879
119880 (0) = [1199061(0) 1199062(0) 119906
119873minus1(0)]119879
119891 = [119891(1199091) 119891(119909
2) 119891(119909
119873minus1)]119879
119865 (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865
2(119905 119906 (119905)) 119865
119873minus1(119905 119906 (119905))]
119879
119865119899(119905 119906 (119905)) = minus ]119906120575
119899(119905)
119873minus1
sum
119894=1
119860119899119894119906119894(119905) +
119873minus1
sum
119894=1
119861119899119894119906119894(119905)
minus ]119906120575119899(119905) 119889119899(119905) +
119889119899(119905) + 120574119906
119899(119905) (1 minus 119906
120575
119899(119905))
119899 = 1 119873 minus 1
(25)
Then (24) can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(26)
This system of ordinary differential equations can be solvedby using four-stage A-stable implicit Runge-Kutta scheme
32 (1+1)-Dimensional Fisher Equation with Variable Coef-ficient In this subsection we extend the application of theLegendre pseudospectral method to solve numerically theFisher equation with variable coefficient
119906119905minus 119887 (119905) 119906
119909119909minus 119888119906 (1 minus 119906) = 0 (119909 119905) isin 119863 times [0 119879] (27)
subject to the initial-boundary conditions
119906 (119909 0) = 119891 (119909) 119909 isin 119863
119906 (119909 119905) = 119892 (119905)
(28)
Proceeding as in the previous subsection we can obtain 119906119909119909
in the same form as (19) and then (27) can be collocated inthe Legendre Gauss-Lobatto points as
119906sdot
119899(119905) minus 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) minus 119887 (119905)
119889119899(119905) minus 119888119906
119899(119905) (1 minus 119906
119899(119905))
= 0 119899 = 1 119873 minus 1
(29)
Table 1 Absolute errors for Example 1
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
556 times 10minus11
minus1
02
893 times 10minus11
minus05 146 times 10minus8
minus05 823 times 10minus9
0 195 times 10minus8 0 141 times 10
minus8
05 155 times 10minus8 05 117 times 10
minus8
1 556 times 10minus11 1 893 times 10
minus11
which can be written in the matrix form
119880sdot(119905) = 119865 (119905 119906 (119905))
119880 (0) = 119891
(30)
where
119865119899(119905 119906 (119905))
= 119887 (119905)
119873minus1
sum
119894=1
119861119899119894119906119894(119905) + 119887 (119905)
119889119899(119905) + 119888119906
119899(119905) (1 minus 119906
119899(119905))
119899 = 1 119873 minus 1
(31)
4 Numerical Examples
In this section three nonlinear time-dependent Fisher-typeequations on finite interval are implemented to demonstratethe accuracy and capability of the proposed algorithm and allof them were performed on the computer using a programwritten in Mathematica 80 The absolute errors in thegiven tables are 119864(119909 119905) = |119906(119909 119905) minus (119909 119905) where 119906(119909 119905)
and (119909 119905) are the exact and numerical solution at selectedpoints (119909 119905)
Example 1 Consider the nonlinear time-dependent one-dimensional Fisher-type equations
119906119905= 119906119909119909
+ 119906 (1 minus 119906) (119906 minus 120574) (119909 119905) isin 119863 times [0 119879] (32)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 minus
1 + 120574
radic2
119905)]
119906 (minus1 119905) =
1 + 120574
2
+
120574 minus 1
2
tanh [120574 minus 1
2radic2
(1 +
1 + 120574
radic2
119905)]
119906 (119909 0) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909)] 119909 isin 119863
(33)
The exact solution is
119906 (119909 119905) =
1 + 120574
2
minus
120574 minus 1
2
tanh [120574 minus 1
2radic2
(119909 minus
1 + 120574
radic2
119905)] (34)
In Table 1 we introduce the absolute errors between theapproximate and exact solutions for problem (32) using theproposed method for different values of 119909 and 119905 with 120574 =
10minus2 and 119873 = 20
Abstract and Applied Analysis 5
07
06
05
04
minus10
minus05
00
05
10
10
05
00
u
t
x
(a)
00
05
10
t
x
E
4
times10minus8
3
2
1
0
minus10
minus05
00
05
10
(b)
Figure 1 The result of the L-GL-C method at 120574 = 10minus2 and 119873 = 20 (a) The approximate solution (b) The absolute error
100500
05
minus10
04
minus05
06
07
08
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 2 The curves of approximate solutions and the exactsolutions of problem (32) at 119905 = 00 119905 = 05 and 119905 = 09 with 120574 =
10minus2 and 119873 = 20
In case of 120574 = 10minus2 and 119873 = 20 the approximate solu-
tion and absolute errors of problem (32) are displayed inFigures 1(a) and 1(b) respectively In Figure 2 we plotted thecurves of approximate solutions and exact solutions of pro-blem (32) for different values of 119905 (0005 and 09) with 120574 =
10minus2 and119873 = 20 It is clear from this figure that approximate
solutions and exact solutions completely coincide for thechosen values of 119905
Example 2 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equations
119906119905= 119906119909119909
minus ]119906120575119906119909+ 120574119906 (1 minus 119906
120575) (119909 119905) isin 119863 times [0 119879]
(35)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905)
= (
1
2
minus
1
2
tanh [
]120575
2 (120575 + 1)
times (1 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (minus1 119905)
= (
1
2
+
1
2
tanh [ ]120575
2 (120575 + 1)
times (1 + (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (119909 0) = (
1
2
minus
1
2
tanh [ ]120575
2 (120575 + 1)
119909])
1120575
119909 isin 119863
(36)
6 Abstract and Applied Analysis
0502
minus10
minus05
00
05
10 00
05
10
x
u
t
0500
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
8
times10minus10
6
4
2
0
(b)
Figure 3 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 1 and 119873 = 20 (a) The approximate solution (b)The absolute error
05005
05010
05015
05020
05025
05030
100500minus10 minus05
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 4 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 1 and119873 = 20
The exact solution of (35) is
119906 (119909 119905)
= (
1
2
minus
1
2
tanh( ]120575
2 (120575 + 1)
times(119909 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)))
1120575
(37)
The absolute errors for problem (35) are listed in Table 2using the L-GL-C method with ] = 120574 = 10
minus2 119873 = 20 andvarious choices of 120575
To illustrate the effectiveness of the Legendre pseudospec-tralmethod for problem (35) we displayed in Figures 3(a) and3(b) the approximate solution and the absolute error with ] =
120574 = 10minus2 120575 = 1 and 119873 = 20 The graph of curves of exact
and approximate solutions with different values of 119905 (0005 and 09) is given in Figure 4 Moreover the approximatesolution and the absolute error with ] = 120574 = 10
minus2 120575 = 2 and119873 = 20 are displayed in Figures 5(a) and 5(b) respectivelyThe curves of exact and approximate solutions of problem(35) with 120575 = 2 are displayed in Figure 6 with values ofparameters listed in its caption
Example 3 Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient
119906119905= minus
119886
61205832coth(119886
6
119905 + 119888) 119906119909119909
+ 119886119906 (1 minus 119906)
(119909 119905) isin 119863 times [0 119879]
(38)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583
2
+
5119886
12
119905)
+
1
2
tanh(120583
2
+
5119886
12
119905) +
1
2
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
07
06
05
04
minus10
minus05
00
05
10
10
05
00
u
t
x
(a)
00
05
10
t
x
E
4
times10minus8
3
2
1
0
minus10
minus05
00
05
10
(b)
Figure 1 The result of the L-GL-C method at 120574 = 10minus2 and 119873 = 20 (a) The approximate solution (b) The absolute error
100500
05
minus10
04
minus05
06
07
08
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 2 The curves of approximate solutions and the exactsolutions of problem (32) at 119905 = 00 119905 = 05 and 119905 = 09 with 120574 =
10minus2 and 119873 = 20
In case of 120574 = 10minus2 and 119873 = 20 the approximate solu-
tion and absolute errors of problem (32) are displayed inFigures 1(a) and 1(b) respectively In Figure 2 we plotted thecurves of approximate solutions and exact solutions of pro-blem (32) for different values of 119905 (0005 and 09) with 120574 =
10minus2 and119873 = 20 It is clear from this figure that approximate
solutions and exact solutions completely coincide for thechosen values of 119905
Example 2 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equations
119906119905= 119906119909119909
minus ]119906120575119906119909+ 120574119906 (1 minus 119906
120575) (119909 119905) isin 119863 times [0 119879]
(35)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905)
= (
1
2
minus
1
2
tanh [
]120575
2 (120575 + 1)
times (1 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (minus1 119905)
= (
1
2
+
1
2
tanh [ ]120575
2 (120575 + 1)
times (1 + (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)])
1120575
119906 (119909 0) = (
1
2
minus
1
2
tanh [ ]120575
2 (120575 + 1)
119909])
1120575
119909 isin 119863
(36)
6 Abstract and Applied Analysis
0502
minus10
minus05
00
05
10 00
05
10
x
u
t
0500
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
8
times10minus10
6
4
2
0
(b)
Figure 3 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 1 and 119873 = 20 (a) The approximate solution (b)The absolute error
05005
05010
05015
05020
05025
05030
100500minus10 minus05
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 4 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 1 and119873 = 20
The exact solution of (35) is
119906 (119909 119905)
= (
1
2
minus
1
2
tanh( ]120575
2 (120575 + 1)
times(119909 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)))
1120575
(37)
The absolute errors for problem (35) are listed in Table 2using the L-GL-C method with ] = 120574 = 10
minus2 119873 = 20 andvarious choices of 120575
To illustrate the effectiveness of the Legendre pseudospec-tralmethod for problem (35) we displayed in Figures 3(a) and3(b) the approximate solution and the absolute error with ] =
120574 = 10minus2 120575 = 1 and 119873 = 20 The graph of curves of exact
and approximate solutions with different values of 119905 (0005 and 09) is given in Figure 4 Moreover the approximatesolution and the absolute error with ] = 120574 = 10
minus2 120575 = 2 and119873 = 20 are displayed in Figures 5(a) and 5(b) respectivelyThe curves of exact and approximate solutions of problem(35) with 120575 = 2 are displayed in Figure 6 with values ofparameters listed in its caption
Example 3 Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient
119906119905= minus
119886
61205832coth(119886
6
119905 + 119888) 119906119909119909
+ 119886119906 (1 minus 119906)
(119909 119905) isin 119863 times [0 119879]
(38)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583
2
+
5119886
12
119905)
+
1
2
tanh(120583
2
+
5119886
12
119905) +
1
2
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
0502
minus10
minus05
00
05
10 00
05
10
x
u
t
0500
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
8
times10minus10
6
4
2
0
(b)
Figure 3 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 1 and 119873 = 20 (a) The approximate solution (b)The absolute error
05005
05010
05015
05020
05025
05030
100500minus10 minus05
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 4 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 1 and119873 = 20
The exact solution of (35) is
119906 (119909 119905)
= (
1
2
minus
1
2
tanh( ]120575
2 (120575 + 1)
times(119909 minus (
]
120575 + 1
+
120574 (120575 + 1)
]) 119905)))
1120575
(37)
The absolute errors for problem (35) are listed in Table 2using the L-GL-C method with ] = 120574 = 10
minus2 119873 = 20 andvarious choices of 120575
To illustrate the effectiveness of the Legendre pseudospec-tralmethod for problem (35) we displayed in Figures 3(a) and3(b) the approximate solution and the absolute error with ] =
120574 = 10minus2 120575 = 1 and 119873 = 20 The graph of curves of exact
and approximate solutions with different values of 119905 (0005 and 09) is given in Figure 4 Moreover the approximatesolution and the absolute error with ] = 120574 = 10
minus2 120575 = 2 and119873 = 20 are displayed in Figures 5(a) and 5(b) respectivelyThe curves of exact and approximate solutions of problem(35) with 120575 = 2 are displayed in Figure 6 with values ofparameters listed in its caption
Example 3 Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient
119906119905= minus
119886
61205832coth(119886
6
119905 + 119888) 119906119909119909
+ 119886119906 (1 minus 119906)
(119909 119905) isin 119863 times [0 119879]
(38)
where 119863 = 119909 minus1 lt 119909 lt 1 Subject to
119906 (1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583
2
+
5119886
12
119905)
+
1
2
tanh(120583
2
+
5119886
12
119905) +
1
2
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
minus10
minus05
00
05
1000
05
10
t
x
0710
0708
0706
u
(a)
minus10
minus05
00
05
10 00
05
10
t
x
E
times10minus8
1
2
0
(b)
Figure 5 The result of the L-GL-C method at ] = 120574 = 10minus2 120575 = 2 and 119873 = 20 (a) The approximate solution (b)The absolute error
Table 2 Absolute errors for Example 2
120575 = 1 120575 = 2 120575 = 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
851 times 10minus11
minus1
01
126 times 10minus10
minus1
01
691 times 10minus11
minus05 116 times 10minus10
minus05 234 times 10minus8
minus05 109 times 10minus8
0 536 times 10minus12 0 261 times 10
minus8 0 130 times 10minus8
05 153 times 10minus11 05 233 times 10
minus8 05 109 times 10minus8
1 851 times 10minus11 1 126 times 10
minus10 1 691 times 10minus11
minus1
05
871 times 10minus11
minus1
05
122 times 10minus10
minus1
05
141 times 10minus10
minus05 777 times 10minus10
minus05 824 times 10minus9
minus05 427 times 10minus9
0 649 times 10minus10 0 116 times 10
minus8 0 589 times 10minus9
05 381 times 10minus10 05 826 times 10
minus9 05 428 times 10minus9
1 871 times 10minus11 1 122 times 10
minus10 1 141 times 10minus10
119906 (minus1 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (minus120583
2
+
5119886
12
119905)
+
1
2
tanh(minus120583
2
+
5119886
12
119905) +
1
2
(39)
119906 (119909 0) =
1
4
coth (119888) sech2 (120583119909
2
)
+
1
2
tanh(120583119909
2
) +
1
2
119909 isin 119863
(40)
The exact solution of (38) is
119906 (119909 119905) =
1
4
coth(1198866
119905 + 119888) sech2 (120583119909
2
+
5119886
12
119905)
+
1
2
tanh(120583119909
2
+
5119886
12
119905) +
1
2
(41)
Table 3 lists the absolute errors for problem (38) usingthe L-GL-C method From numerical results of this table itcan be concluded that the numerical solutions are in excellentagreement with the exact solutions
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
minus10 minus05 00 05 10
0707
0708
0709
0710
0711
x
uandu
u(x 0)
u(x 0)
u(x 05)
u(x 05)
u(x 09)
u(x 09)
Figure 6 The curves of approximate solutions and the exactsolutions of problem (35) at 119905 = 00 119905 = 05 and 119905 = 09 with] = 120574 = 10
minus2 120575 = 2 and 119873 = 20
Table 3 Absolute errors for Example 3
119909 119905 119864(119909 119905) 119909 119905 119864(119909 119905)
minus1
01
981 times 10minus10
minus1
05
705 times 10minus9
minus05 100 times 10minus7
minus05 193 times 10minus6
0 145 times 10minus7 0 284 times 10
minus6
05 119 times 10minus7 05 212 times 10
minus6
1 981 times 10minus10 1 706 times 10
minus9
5 Conclusion
In this paper based on the Legendre-Gauss-Lobatto pseudos-pectral approximation we proposed an efficient numericalalgorithm to solve nonlinear time-dependent Fisher-typeequationswith constant and variable coefficientsThemethodis based upon reducing the nonlinear partial differentialequation into a system of first-order ordinary differentialequations in the expansion coefficient of the spectral solutionNumerical examples were also provided to illustrate theeffectiveness of the derived algorithmsThenumerical experi-ments show that the Legendre pseudospectral approximationis simple and accurate with a limited number of collocationnodes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] D A Kopriva Implementing Spectral Methods for Partial Dif-ferential Equations Algorithms for Scientists and EngineersSpringer Berlin Germany 2009
[2] C Canuto M Y Hussaini A Quarteroni and T A Zang Spec-tral Methods Fundamentals in Single Domains Springer BerlinGermany 2006
[3] C I Gheorghiu Spectral Methods for Differential ProblemsT Popoviciu Institute of Numerical Analysis Cluj-NapocaRomaina 2007
[4] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013
[5] EHDoha andAH Bhrawy ldquoAn efficient direct solver formul-tidimensional elliptic Robin boundary value problems using aLegendre spectral-Galerkin methodrdquo Computers amp Mathema-tics with Applications vol 64 no 4 pp 558ndash571 2012
[6] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013
[7] O R Isik and M Sezer ldquoBernstein series solution of a class ofLane-Emden type equationsrdquo Mathematical Problems in Engi-neering Article ID 423797 9 pages 2013
[8] E Tohidi A H Bhrawy and K Erfani ldquoA collocation methodbased on Bernoulli operationalmatrix for numerical solution ofgeneralized pantograph equationrdquo Applied Mathematical Mod-elling vol 37 no 6 pp 4283ndash4294 2013
[9] M S Mechee and N Senu ldquoNumerical study of fractional dif-ferential equations of Lane-Emden type by method of colloca-tionrdquo Applied Mathematics vol 3 pp 851ndash856 2012
[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011
[11] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013
[12] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficient gen-eralized Laguerre spectral algorithms for fractional initial valueproblemsrdquo Abstract and Applied Analysis vol 2013 Article ID546502 10 pages 2013
[13] D Baleanu A H Bhrawy and T M Taha ldquoA modified Gen-eralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013
[14] A Ahmadian M Suleiman and S Salahshour ldquoAn opera-tional matrix based on Legendre polynomials for solving fuzzyfractional-order differential equationsrdquo Abstract and AppliedAnalysis vol 2013 Article ID 505903 29 pages 2013
[15] F Ghaemi R Yunus A Ahmadian S Salahshourd MSuleiman and S F Saleh ldquoApplication of fuzzy fractional kineticequations tomodelling of the acid hydrolysis reactionrdquoAbstractand Applied Analysis vol 2013 Article ID 610314 19 pages 2013
[16] M H Atabakzadeh M H Akrami and G H Erjaee ldquoCheby-shev operational matrix method for solving multi-order frac-tional ordinary differential equationsrdquo Applied MathematicalModelling vol 37 no 20-21 pp 8903ndash8911 2013
[17] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 19 pages 2012
[18] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling 2013
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
[19] M R Eslahchi M Dehghan and M Parvizi ldquoApplication ofthe collocationmethod for solving nonlinear fractional integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 257 pp 105ndash128 2013
[20] Y Yangy and Y Huang ldquoSpectral-collocation methods forfractional pantograph delay-integrodifferential equationsrdquoAdvances in Mathematical Physics In press
[21] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[22] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937
[23] A J Khattak ldquoA computational meshless method for the gen-eralized Burgerrsquos-Huxley equationrdquoAppliedMathematicalMod-elling vol 33 no 9 pp 3718ndash3729 2009
[24] N F BrittonReactiondiffusion Equations andTheir Applicationsto Biology Academic Press London UK 1986
[25] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955
[26] A Wazwaz ldquoThe extended tanh method for abundant solitarywave solutions of nonlinear wave equationsrdquo Applied Mathe-matics and Computation vol 187 no 2 pp 1131ndash1142 2007
[27] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992
[28] Y Tan H Xu and S-J Liao ldquoExplicit series solution of travel-ling waves with a front of Fisher equationrdquo Chaos Solitons andFractals vol 31 no 2 pp 462ndash472 2007
[29] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004
[30] J Canosa ldquoDiffusion in nonlinearmultiplicativemediardquo Journalof Mathematical Physics vol 10 no 10 pp 1862ndash1868 1969
[31] A Ogun and C Kart ldquoExact solutions of Fisher and generalizedFisher equations with variable coefficientsrdquo Acta MathematicaeApplicatae Sinica vol 23 no 4 pp 563ndash568 2007
[32] M D Gunzburger L S Hou and W Zhu ldquoFully discrete finiteelement approximations of the forced Fisher equationrdquo Journalof Mathematical Analysis and Applications vol 313 no 2 pp419ndash440 2006
[33] I Dag A Sahin and A Korkmaz ldquoNumerical investigationof the solution of Fisherrsquos equation via the B-spline galerkinmethodrdquo Numerical Methods for Partial Differential Equationsvol 26 no 6 pp 1483ndash1503 2010
[34] M Bastani and D K Salkuyeh ldquoA highly accurate method tosolve Fisherrsquos equationrdquo Pramana vol 78 no 3 pp 335ndash3462012
[35] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearFisherrsquos reaction-diffusion equation with modified cubic B-spline collocationmethodrdquoMathematical Sciences vol 7 article12 2013
[36] J Gazdag and J Canosa ldquoNumerical solution of Fisherrsquos equa-tionrdquo Journal of Applied Probability vol 11 pp 445ndash457 1974
[37] T Zhao C Li Z Zang and Y Wu ldquoChebyshev-Legendrepseudo-spectral method for the generalised Burgers-Fisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012
[38] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquo
AppliedMathematics and Computation vol 216 no 8 pp 2472ndash2478 2010
[39] X Y Chen Numerical methods for the Burgers-Fisher equation[MS thesis] University of Aeronautics and Astronautics Nan-jing China 2007
[40] R E Mickens and A B Gumel ldquoConstruction and analysis ofa non-standard finite difference scheme for the Burgers-Fisherequationrdquo Journal of Sound and Vibration vol 257 no 4 pp791ndash797 2002
[41] R E Mickens ldquoA best finite-difference scheme for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 10 no 5 pp 581ndash585 1994
[42] R E Mickens ldquoRelation between the time and space step-sizes in nonstandard finite-difference schemes for the FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 15 no 1 pp 51ndash55 1997
[43] N Parekh and S Puri ldquoA new numerical scheme for the Fisherequationrdquo Journal of Physics A vol 23 no 21 pp L1085ndashL10911990
[44] R Rizwan-Uddin ldquoComparison of the nodal integral methodand nonstandard finite-difference schemes for the Fisher equa-tionrdquo SIAM Journal on Scientific Computing vol 22 no 6 pp1926ndash1942 2001
[45] R Chernma ldquoExact and numerical solutions of tiie generalizedfisher equationrdquo Reports on Mathematkxl Physics vol 47 pp393ndash411 2001
[46] A Sahin I Dag and B Saka ldquoA B-spline algorithm for thenumerical solution of Fisherrsquos equationrdquo Kybernetes vol 37 no2 pp 326ndash342 2008
[47] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[48] J Roessler and H Hussner ldquoNumerical solution of the 1 +
2 dimensional Fisherrsquos equation by finite elements and theGalerkin methodrdquoMathematical and Computer Modelling vol25 no 3 pp 57ndash67 1997
[49] E H Doha A H Bhrawy R M Hafez and M A AbdelkawyldquoA Chebyshev-Gauss-Radau scheme for nonlinear hyperbolicsystem of first orderrdquo Applied Mathematics and InformationScience vol 8 no 2 pp 1ndash10 2014
[50] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013
[51] E H Doha D Baleanu A H Bhrawy andM A Abdelkawy ldquoAJacobi collocation method for solving nonlinear Burgersrsquo-typeequationsrdquo Abstract and Applied Analysis vol 2013 Article ID760542 12 pages 2013
[52] A H Bhrawy L M Assas and M A Alghamdi ldquoFast spec-tral collocationmethod for solving nonlinear time-delayed Bur-gers-type equations with positive power termsrdquo Abstract andApplied Analysis vol 2013 Article ID 741278 12 pages 2013
[53] A H Bhrawy L M Assas and M A Alghamdi ldquoAn efficientspectral collocation algorithm for nonlinear Phi-four equa-tionsrdquo Boundary Value Problems vol 2013 article 87 16 pages2013
[54] A Saadatmandi ldquoBernstein operational matrix of fractionalderivatives and its applicationsrdquo Applied Mathematical Mod-elling 2013
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
[55] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Related Documents
