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Research ArticleA Reproducing Kernel Hilbert Space Method for SolvingSystems of Fractional Integrodifferential Equations
Samia Bushnaq1 Banan Maayah2 Shaher Momani23 and Ahmed Alsaedi3
1 Department of Science King Abdullah II Faculty of Engineering Princess Sumaya University for Technology Amman 11941 Jordan2Department of Mathematics Faculty of Science The University of Jordan Amman 11942 Jordan3Nonlinear Analysis and Applied Mathematics (NAAM) Research Group Faculty of Science King Abdulaziz UniversityJeddah 21589 Saudi Arabia
Correspondence should be addressed to Shaher Momani smomanijuedujo
Received 5 January 2014 Accepted 14 February 2014 Published 24 March 2014
Academic Editor Hossein Jafari
Copyright copy 2014 Samia Bushnaq et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present a new version of the reproducing kernel Hilbert space method (RKHSM) for the solution of systems of fractionalintegrodifferential equations In this approach the solution is obtained as a convergent series with easily computable componentsSeveral illustrative examples are given to demonstrate the effectiveness of the present method The method described in this paperis expected to be further employed to solve similar nonlinear problems in fractional calculus
1 Introduction
In this paper we consider the following system of fractionalintegrodifferential equations
119863120572119894119909119894(119905)
= 119865119894(119905 1199091(119905) 119909
(119896)
1(119905) 119909
119894minus1(119905) 119909
(119896)
119894minus1(119905)
119909119894+1(119905) 119909
(119896)
119894+1(119905) 119909
119899(119905) 119909
(119896)
119899(119905))
+ int
119905
0
119866119894(119905 120591 119909
1(120591) 119909
(119896)
1(120591) 119909
119899(120591)
119909(119896)
119899(120591)) 119889120591
(1)
where 119894 = 1 119899 119896 = 0 1 119898 0 le 119905 le 1 and 119863120572119894 isderivative of order 120572
119894in the sense of Caputo and119898minus1 lt 120572
119894le
119898 subject to the initial conditions
119909(119895)
119894(119886) = 119886
119895119894 119895 = 0 1 119898 minus 1 119894 = 1 2 119899 119886 ge 0
(2)
In the last two decades fractional calculus has founddiverse applications in various scientific and technologi-cal fields [1 2] such as thermal engineering acoustics
electromagnetism control robotics viscoelasticity diffu-sion edge detection turbulence signal processing andmany other physical and biological processes Fractional dif-ferential equations have also been applied in modelingmany physical and engineering problems Most systems offractional integrodifferential equations do not have exactsolutions so numerical techniques are used to solve suchsystems The homotopy perturbation method the Adomiandecomposition method and other methods are used to givean approximate solution to linear and nonlinear problemssee [3ndash13] and the references therein
In our previous work [14] we proposed a reproducingkernel Hilbert space method for solving integrodifferentialequations of fractional order based on the reproducing kerneltheory [14 15] In this paper we will generalize the idea ofthe RKHSM to provide a numerical solution for systems offractional integrodifferential equations (1) To demonstratethe effectiveness of the RKHSM algorithm several numericalexperiments of linear and nonlinear systems of fractionalequations (1) will be presented
This paper is organized as follows An introduction of thealgorithm for solving systems of fractional integrodifferentialequations is given in Section 2 In Section 3 we introduceseveral examples to show the efficiency of themethod Finallya conclusion is given in Section 4
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 103016 6 pageshttpdxdoiorg1011552014103016
2 Abstract and Applied Analysis
2 The Algorithm
After homogenizing the initial conditions (2) we apply theoperator 119868120572119894 the Riemann-Liouville fractional integral oforder 120572
It is clear that (3) is equivalent to (1) so every solutionof the integral equation (3) is also a solution of our originalproblem (1) and vice versa
To solve (3) by means of the reproducing kernel Hilbertspace method first we need to construct a reproducingkernel of certain spaces119882119898+1
2[119886 119887] = 119906 | 119906
(119895) is absolutelycontinuous 119895 = 1 2 119898 minus 1 and 119906(119898) isin 1198712[119886 119887] in whichevery function satisfies the homogenous initial conditions of(1)
(i) The inner product of the space 11988212[0 1] = 119906 | 119906
is absolutely continuous real value function 1199061015840 isin1198712
reproducing kernel Hilbert space and its reproducingkernel is given by
119872(119909 119910) = 119891 (119909 119910) 119910 le 119909
119891 (119910 119909) 119910 gt 119909(10)
where 119891(119909 119910) = (1120)1199102(minus1199092(minus126 + 10119909 minus 51199092 + 1199093) +5(minus1 + 119909)119909119910
2
minus (minus1 + 1199092
)1199103
)The method of obtaining the reproducing kernel can be
found in [15]Let 119871
119894 119882119898+1
2[0 1] rarr 119882
1
2[0 1] such that 119871
119894119909119894(119905) =
119909119894(119905) Then 119871
119894 119894 = 1 2 119899 are bounded linear operators
Let 119905119895infin
119895=1
be a countable dense set in [0 1] Let 120593119894119895(119905) =
119877(119905119895 119905) and 120595119894
119895(119905) = 119871
lowast
119894120593119894
119895(119905) where 119871lowast
119894is the adjoint operator
of 119871119894By Gram-Schmidt process we can construct an orthonor-
mal system 120595119894119895(119905)infin
119895=1
of119882119898+12[0 1] where
120595119894
119895(119905) =
119895
sum
119896=1
120573119894
119895119896120595119894
119896(119905) 120573
119894
119895119895gt 0
forall119895 = 1 2 119894 = 1 2 119899
(11)
Theorem 1 Let 119905119895infin
119895=1be a dense set in [0 1] Then 120595119894
119895(119905)infin
119895=1
is a complete system of119882119898+12[0 1]
For the proof see [14]
Theorem 2 Let 119905119895infin
119895=1be a dense set in [0 1] and the solution
of (3) is unique on119882119898+12[0 1]Then the solution of (3) is given
by 119909119894(119905) = sum
infin
119895=1119860119895120595119894
119895(119905) where 119860
119895= sum119895
119896=1120573119894
119895119896119872119894(119905119896)
For the proof see [14]One can get an approximate solution 119909
119894119899(119905) by taking
finitely many terms in the series representation of 119909119894(119905) and
119909119894119899(119905) = sum
119899
119895=1119860119895120595119894
119895(119905)
Since 119882119898+1
2[0 1] is a Hilbert space then
suminfin
119895=1suminfin
119896=1120573119894
119895119896119872119894(119905119896) lt infin
Abstract and Applied Analysis 3
02 04 06 08 10
15
20
25
30
35
40x(t)
t
120572 = 1
120572 = 09
120572 = 08
120572 = 07
(a)
02 04 06 08 10
000005
000010
000015
000020
000025
000030
Abso
lute
erro
r
t
(b)
02 04 06 08 10t
y(t)
minus05
minus10
minus15
120572 = 1
120572 = 09
120572 = 08
120572 = 07
(c)
Abso
lute
erro
r
02 04 06 08 10
000005
000010
000015
000020
000025
000030
t
(d)
Figure 1 Graphical results for Example 1 when 1205721= 1205722= 120572 = 1 09 08 and 07
Theorem 3 The approximate solution 119909119894119899(119905) and its deriva-
tives 119909(119895)119894119899
are uniformly convergent to 119909(119895)119894(119905) 119894 = 1 2 119899
119895 = 0 1
Proof By the reproducing kernel property of 119870(119909 119910) andSchwarz inequality we can obtain
1003816100381610038161003816119909119894119899 (119905) minus 119909119894 (119905)1003816100381610038161003816 =100381610038161003816100381610038161003816⟨119909119894119899(119905) minus 119909
Thus 119909119894(119905) and its derivatives 119909(119895)
119894119899(119905) are uniformly convergent
to 119909(119895)119894(119905) 119895 = 1 2
3 Numerical Results
In this paper three numerical examples are given to show theaccuracy of this methodThe computations are performed byMathematica 80We compare the results by thismethodwiththe exact solution of each example
Example 1 Consider the following linear system of fractionalintegrodifferential equations
The exact solution for 1205721= 1205722= 1 is 119909(119905) = sinh 119905 119910(119905) =
cosh 119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 30 the graphs
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 2
Example 3 Consider the following nonlinear system offractional integrodifferential equations
1198631205721119909 (119905) = 1 minus
1199053
3minus1
211991010158402
(119905) +1
2int
119905
0
(1199092
(120591) + 1199102
(120591)) 119889120591
1198631205722119909 (119905) = minus1 + 119905
2
minus 119905119909 (119905) +1
4int
119905
0
(1199092
(120591) minus 1199102
(120591)) 119889120591
119909 (0) = 1 1199091015840
(0) = 2 119910 (0) = minus1
1199101015840
(0) = 0 1 lt 1205721 1205722le 2
(18)
The exact solution for 1205721= 1205722= 2 is 119909(119905) = 119905 + 1198903 119910(119905) =
119905 minus 119890119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 20 the graphs
6 Abstract and Applied Analysis
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 3
4 Conclusion
In this paper we introduce a new algorithm for solving sys-tems of fractional integrodifferential equations The approxi-mate solution obtained by this method and its derivative areboth uniformly convergentThe obtained results demonstratethe reliability of the algorithm and its wider applicabilityto linear and nonlinear systems of fractional differentialequations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Podlubuy Fractional Differential Equations Academic PressNew York NY USA 1999
[2] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanics rdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds Springer New York NY USA 1997
[3] V S Erturk and S Momani ldquoSolving systems of fractionaldifferential equations using differential transform methodrdquoJournal of Computational and AppliedMathematics vol 215 no1 pp 142ndash151 2008
[4] J Duan J An and M Xu ldquoSolution of system of fractionaldifferential equations by Adomian decomposition methodrdquoApplied Mathematics vol 22 no 1 pp 7ndash12 2007
[5] A S Mohamed and R A Mahmoud ldquoAn algorithm for thenumerical solution of systems of fractional differential equa-tionsrdquo International Journal of Computer Applications vol 65no 11 2013
[6] S Momani and R Qaralleh ldquoAn efficient method for solvingsystems of fractional integro-differential equationsrdquo Computersamp Mathematics with Applications vol 52 no 3-4 pp 459ndash4702006
[7] M Zurigat S Momani and A Alawneh ldquoHomotopy analysismethod for systems of fractional integro-differential equationsrdquoin Proceeding of the 4th International Workshop of AdvancedComputation for Engineering Applications pp 106ndash111 2008
[8] M Zurigat SMomani Z Odibat andA Alawneh ldquoThe homo-topy analysis method for handling systems of fractional differ-ential equationsrdquo Applied Mathematical Modelling vol 34 no1 pp 24ndash35 2010
[9] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007
[10] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 4 pp 451ndash459 2008
[11] Y Lin and Y Zhou ldquoSolving nonlinear pseudoparabolic equa-tions with nonlocal boundary conditions in reproducing kernelspacerdquo Numerical Algorithms vol 52 no 2 pp 173ndash186 2009
[12] B Ghazanfari and A G Ghazanfari ldquoSolving system of frac-tional differential equations by fractional complex transformmethodrdquoAsian Journal of Applied Sciences vol 5 no 6 pp 438ndash444 2012
[13] R K Saeed and H M Sadeq ldquoSolving a system of linear fred-holm fractional integro-differential equations using homotopyperturbation methodrdquo Australian Journal of Basic and AppliedSciences vol 4 no 4 pp 633ndash638 2010
[14] S Bushnaq S Momani and Y Zhon ldquoA reproducing KernelHilbert space method for solving integro-differential equationsof fractional orderrdquo Journal of Optimization Theory and Appli-cations vol 156 no 1 pp 96ndash105 2012
[15] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Hilbert Space Nova Science Publishers NewYork NY USA 2009
[16] J S Leszczynski An Introduction to Fractional MechanicsCzestochowa University of Technology Czestochowa Poland2011
[17] M Caputo ldquoLinear Models of Dissipation whose Q is almostFrequency Independent-IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967
[18] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006
[19] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000
[20] S G Samko A A Kilbas andO IMarichev Fractional Integraland Derivatives Theory and Applications Gordon and BreachScience Publishers Yverdon Switzerland 1993
[21] C-l Li and M-G Cui ldquoThe exact solution for solving a classnonlinear operator equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 143 no 2-3 pp393ndash399 2003
It is clear that (3) is equivalent to (1) so every solutionof the integral equation (3) is also a solution of our originalproblem (1) and vice versa
To solve (3) by means of the reproducing kernel Hilbertspace method first we need to construct a reproducingkernel of certain spaces119882119898+1
2[119886 119887] = 119906 | 119906
(119895) is absolutelycontinuous 119895 = 1 2 119898 minus 1 and 119906(119898) isin 1198712[119886 119887] in whichevery function satisfies the homogenous initial conditions of(1)
(i) The inner product of the space 11988212[0 1] = 119906 | 119906
is absolutely continuous real value function 1199061015840 isin1198712
reproducing kernel Hilbert space and its reproducingkernel is given by
119872(119909 119910) = 119891 (119909 119910) 119910 le 119909
119891 (119910 119909) 119910 gt 119909(10)
where 119891(119909 119910) = (1120)1199102(minus1199092(minus126 + 10119909 minus 51199092 + 1199093) +5(minus1 + 119909)119909119910
2
minus (minus1 + 1199092
)1199103
)The method of obtaining the reproducing kernel can be
found in [15]Let 119871
119894 119882119898+1
2[0 1] rarr 119882
1
2[0 1] such that 119871
119894119909119894(119905) =
119909119894(119905) Then 119871
119894 119894 = 1 2 119899 are bounded linear operators
Let 119905119895infin
119895=1
be a countable dense set in [0 1] Let 120593119894119895(119905) =
119877(119905119895 119905) and 120595119894
119895(119905) = 119871
lowast
119894120593119894
119895(119905) where 119871lowast
119894is the adjoint operator
of 119871119894By Gram-Schmidt process we can construct an orthonor-
mal system 120595119894119895(119905)infin
119895=1
of119882119898+12[0 1] where
120595119894
119895(119905) =
119895
sum
119896=1
120573119894
119895119896120595119894
119896(119905) 120573
119894
119895119895gt 0
forall119895 = 1 2 119894 = 1 2 119899
(11)
Theorem 1 Let 119905119895infin
119895=1be a dense set in [0 1] Then 120595119894
119895(119905)infin
119895=1
is a complete system of119882119898+12[0 1]
For the proof see [14]
Theorem 2 Let 119905119895infin
119895=1be a dense set in [0 1] and the solution
of (3) is unique on119882119898+12[0 1]Then the solution of (3) is given
by 119909119894(119905) = sum
infin
119895=1119860119895120595119894
119895(119905) where 119860
119895= sum119895
119896=1120573119894
119895119896119872119894(119905119896)
For the proof see [14]One can get an approximate solution 119909
119894119899(119905) by taking
finitely many terms in the series representation of 119909119894(119905) and
119909119894119899(119905) = sum
119899
119895=1119860119895120595119894
119895(119905)
Since 119882119898+1
2[0 1] is a Hilbert space then
suminfin
119895=1suminfin
119896=1120573119894
119895119896119872119894(119905119896) lt infin
Abstract and Applied Analysis 3
02 04 06 08 10
15
20
25
30
35
40x(t)
t
120572 = 1
120572 = 09
120572 = 08
120572 = 07
(a)
02 04 06 08 10
000005
000010
000015
000020
000025
000030
Abso
lute
erro
r
t
(b)
02 04 06 08 10t
y(t)
minus05
minus10
minus15
120572 = 1
120572 = 09
120572 = 08
120572 = 07
(c)
Abso
lute
erro
r
02 04 06 08 10
000005
000010
000015
000020
000025
000030
t
(d)
Figure 1 Graphical results for Example 1 when 1205721= 1205722= 120572 = 1 09 08 and 07
Theorem 3 The approximate solution 119909119894119899(119905) and its deriva-
tives 119909(119895)119894119899
are uniformly convergent to 119909(119895)119894(119905) 119894 = 1 2 119899
119895 = 0 1
Proof By the reproducing kernel property of 119870(119909 119910) andSchwarz inequality we can obtain
1003816100381610038161003816119909119894119899 (119905) minus 119909119894 (119905)1003816100381610038161003816 =100381610038161003816100381610038161003816⟨119909119894119899(119905) minus 119909
Thus 119909119894(119905) and its derivatives 119909(119895)
119894119899(119905) are uniformly convergent
to 119909(119895)119894(119905) 119895 = 1 2
3 Numerical Results
In this paper three numerical examples are given to show theaccuracy of this methodThe computations are performed byMathematica 80We compare the results by thismethodwiththe exact solution of each example
Example 1 Consider the following linear system of fractionalintegrodifferential equations
The exact solution for 1205721= 1205722= 1 is 119909(119905) = sinh 119905 119910(119905) =
cosh 119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 30 the graphs
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 2
Example 3 Consider the following nonlinear system offractional integrodifferential equations
1198631205721119909 (119905) = 1 minus
1199053
3minus1
211991010158402
(119905) +1
2int
119905
0
(1199092
(120591) + 1199102
(120591)) 119889120591
1198631205722119909 (119905) = minus1 + 119905
2
minus 119905119909 (119905) +1
4int
119905
0
(1199092
(120591) minus 1199102
(120591)) 119889120591
119909 (0) = 1 1199091015840
(0) = 2 119910 (0) = minus1
1199101015840
(0) = 0 1 lt 1205721 1205722le 2
(18)
The exact solution for 1205721= 1205722= 2 is 119909(119905) = 119905 + 1198903 119910(119905) =
119905 minus 119890119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 20 the graphs
6 Abstract and Applied Analysis
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 3
4 Conclusion
In this paper we introduce a new algorithm for solving sys-tems of fractional integrodifferential equations The approxi-mate solution obtained by this method and its derivative areboth uniformly convergentThe obtained results demonstratethe reliability of the algorithm and its wider applicabilityto linear and nonlinear systems of fractional differentialequations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Podlubuy Fractional Differential Equations Academic PressNew York NY USA 1999
[2] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanics rdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds Springer New York NY USA 1997
[3] V S Erturk and S Momani ldquoSolving systems of fractionaldifferential equations using differential transform methodrdquoJournal of Computational and AppliedMathematics vol 215 no1 pp 142ndash151 2008
[4] J Duan J An and M Xu ldquoSolution of system of fractionaldifferential equations by Adomian decomposition methodrdquoApplied Mathematics vol 22 no 1 pp 7ndash12 2007
[5] A S Mohamed and R A Mahmoud ldquoAn algorithm for thenumerical solution of systems of fractional differential equa-tionsrdquo International Journal of Computer Applications vol 65no 11 2013
[6] S Momani and R Qaralleh ldquoAn efficient method for solvingsystems of fractional integro-differential equationsrdquo Computersamp Mathematics with Applications vol 52 no 3-4 pp 459ndash4702006
[7] M Zurigat S Momani and A Alawneh ldquoHomotopy analysismethod for systems of fractional integro-differential equationsrdquoin Proceeding of the 4th International Workshop of AdvancedComputation for Engineering Applications pp 106ndash111 2008
[8] M Zurigat SMomani Z Odibat andA Alawneh ldquoThe homo-topy analysis method for handling systems of fractional differ-ential equationsrdquo Applied Mathematical Modelling vol 34 no1 pp 24ndash35 2010
[9] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007
[10] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 4 pp 451ndash459 2008
[11] Y Lin and Y Zhou ldquoSolving nonlinear pseudoparabolic equa-tions with nonlocal boundary conditions in reproducing kernelspacerdquo Numerical Algorithms vol 52 no 2 pp 173ndash186 2009
[12] B Ghazanfari and A G Ghazanfari ldquoSolving system of frac-tional differential equations by fractional complex transformmethodrdquoAsian Journal of Applied Sciences vol 5 no 6 pp 438ndash444 2012
[13] R K Saeed and H M Sadeq ldquoSolving a system of linear fred-holm fractional integro-differential equations using homotopyperturbation methodrdquo Australian Journal of Basic and AppliedSciences vol 4 no 4 pp 633ndash638 2010
[14] S Bushnaq S Momani and Y Zhon ldquoA reproducing KernelHilbert space method for solving integro-differential equationsof fractional orderrdquo Journal of Optimization Theory and Appli-cations vol 156 no 1 pp 96ndash105 2012
[15] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Hilbert Space Nova Science Publishers NewYork NY USA 2009
[16] J S Leszczynski An Introduction to Fractional MechanicsCzestochowa University of Technology Czestochowa Poland2011
[17] M Caputo ldquoLinear Models of Dissipation whose Q is almostFrequency Independent-IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967
[18] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006
[19] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000
[20] S G Samko A A Kilbas andO IMarichev Fractional Integraland Derivatives Theory and Applications Gordon and BreachScience Publishers Yverdon Switzerland 1993
[21] C-l Li and M-G Cui ldquoThe exact solution for solving a classnonlinear operator equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 143 no 2-3 pp393ndash399 2003
Figure 1 Graphical results for Example 1 when 1205721= 1205722= 120572 = 1 09 08 and 07
Theorem 3 The approximate solution 119909119894119899(119905) and its deriva-
tives 119909(119895)119894119899
are uniformly convergent to 119909(119895)119894(119905) 119894 = 1 2 119899
119895 = 0 1
Proof By the reproducing kernel property of 119870(119909 119910) andSchwarz inequality we can obtain
1003816100381610038161003816119909119894119899 (119905) minus 119909119894 (119905)1003816100381610038161003816 =100381610038161003816100381610038161003816⟨119909119894119899(119905) minus 119909
Thus 119909119894(119905) and its derivatives 119909(119895)
119894119899(119905) are uniformly convergent
to 119909(119895)119894(119905) 119895 = 1 2
3 Numerical Results
In this paper three numerical examples are given to show theaccuracy of this methodThe computations are performed byMathematica 80We compare the results by thismethodwiththe exact solution of each example
Example 1 Consider the following linear system of fractionalintegrodifferential equations
The exact solution for 1205721= 1205722= 1 is 119909(119905) = sinh 119905 119910(119905) =
cosh 119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 30 the graphs
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 2
Example 3 Consider the following nonlinear system offractional integrodifferential equations
1198631205721119909 (119905) = 1 minus
1199053
3minus1
211991010158402
(119905) +1
2int
119905
0
(1199092
(120591) + 1199102
(120591)) 119889120591
1198631205722119909 (119905) = minus1 + 119905
2
minus 119905119909 (119905) +1
4int
119905
0
(1199092
(120591) minus 1199102
(120591)) 119889120591
119909 (0) = 1 1199091015840
(0) = 2 119910 (0) = minus1
1199101015840
(0) = 0 1 lt 1205721 1205722le 2
(18)
The exact solution for 1205721= 1205722= 2 is 119909(119905) = 119905 + 1198903 119910(119905) =
119905 minus 119890119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 20 the graphs
6 Abstract and Applied Analysis
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 3
4 Conclusion
In this paper we introduce a new algorithm for solving sys-tems of fractional integrodifferential equations The approxi-mate solution obtained by this method and its derivative areboth uniformly convergentThe obtained results demonstratethe reliability of the algorithm and its wider applicabilityto linear and nonlinear systems of fractional differentialequations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Podlubuy Fractional Differential Equations Academic PressNew York NY USA 1999
[2] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanics rdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds Springer New York NY USA 1997
[3] V S Erturk and S Momani ldquoSolving systems of fractionaldifferential equations using differential transform methodrdquoJournal of Computational and AppliedMathematics vol 215 no1 pp 142ndash151 2008
[4] J Duan J An and M Xu ldquoSolution of system of fractionaldifferential equations by Adomian decomposition methodrdquoApplied Mathematics vol 22 no 1 pp 7ndash12 2007
[5] A S Mohamed and R A Mahmoud ldquoAn algorithm for thenumerical solution of systems of fractional differential equa-tionsrdquo International Journal of Computer Applications vol 65no 11 2013
[6] S Momani and R Qaralleh ldquoAn efficient method for solvingsystems of fractional integro-differential equationsrdquo Computersamp Mathematics with Applications vol 52 no 3-4 pp 459ndash4702006
[7] M Zurigat S Momani and A Alawneh ldquoHomotopy analysismethod for systems of fractional integro-differential equationsrdquoin Proceeding of the 4th International Workshop of AdvancedComputation for Engineering Applications pp 106ndash111 2008
[8] M Zurigat SMomani Z Odibat andA Alawneh ldquoThe homo-topy analysis method for handling systems of fractional differ-ential equationsrdquo Applied Mathematical Modelling vol 34 no1 pp 24ndash35 2010
[9] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007
[10] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 4 pp 451ndash459 2008
[11] Y Lin and Y Zhou ldquoSolving nonlinear pseudoparabolic equa-tions with nonlocal boundary conditions in reproducing kernelspacerdquo Numerical Algorithms vol 52 no 2 pp 173ndash186 2009
[12] B Ghazanfari and A G Ghazanfari ldquoSolving system of frac-tional differential equations by fractional complex transformmethodrdquoAsian Journal of Applied Sciences vol 5 no 6 pp 438ndash444 2012
[13] R K Saeed and H M Sadeq ldquoSolving a system of linear fred-holm fractional integro-differential equations using homotopyperturbation methodrdquo Australian Journal of Basic and AppliedSciences vol 4 no 4 pp 633ndash638 2010
[14] S Bushnaq S Momani and Y Zhon ldquoA reproducing KernelHilbert space method for solving integro-differential equationsof fractional orderrdquo Journal of Optimization Theory and Appli-cations vol 156 no 1 pp 96ndash105 2012
[15] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Hilbert Space Nova Science Publishers NewYork NY USA 2009
[16] J S Leszczynski An Introduction to Fractional MechanicsCzestochowa University of Technology Czestochowa Poland2011
[17] M Caputo ldquoLinear Models of Dissipation whose Q is almostFrequency Independent-IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967
[18] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006
[19] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000
[20] S G Samko A A Kilbas andO IMarichev Fractional Integraland Derivatives Theory and Applications Gordon and BreachScience Publishers Yverdon Switzerland 1993
[21] C-l Li and M-G Cui ldquoThe exact solution for solving a classnonlinear operator equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 143 no 2-3 pp393ndash399 2003
Thus 119909119894(119905) and its derivatives 119909(119895)
119894119899(119905) are uniformly convergent
to 119909(119895)119894(119905) 119895 = 1 2
3 Numerical Results
In this paper three numerical examples are given to show theaccuracy of this methodThe computations are performed byMathematica 80We compare the results by thismethodwiththe exact solution of each example
Example 1 Consider the following linear system of fractionalintegrodifferential equations
The exact solution for 1205721= 1205722= 1 is 119909(119905) = sinh 119905 119910(119905) =
cosh 119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 30 the graphs
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 2
Example 3 Consider the following nonlinear system offractional integrodifferential equations
1198631205721119909 (119905) = 1 minus
1199053
3minus1
211991010158402
(119905) +1
2int
119905
0
(1199092
(120591) + 1199102
(120591)) 119889120591
1198631205722119909 (119905) = minus1 + 119905
2
minus 119905119909 (119905) +1
4int
119905
0
(1199092
(120591) minus 1199102
(120591)) 119889120591
119909 (0) = 1 1199091015840
(0) = 2 119910 (0) = minus1
1199101015840
(0) = 0 1 lt 1205721 1205722le 2
(18)
The exact solution for 1205721= 1205722= 2 is 119909(119905) = 119905 + 1198903 119910(119905) =
119905 minus 119890119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 20 the graphs
6 Abstract and Applied Analysis
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 3
4 Conclusion
In this paper we introduce a new algorithm for solving sys-tems of fractional integrodifferential equations The approxi-mate solution obtained by this method and its derivative areboth uniformly convergentThe obtained results demonstratethe reliability of the algorithm and its wider applicabilityto linear and nonlinear systems of fractional differentialequations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Podlubuy Fractional Differential Equations Academic PressNew York NY USA 1999
[2] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanics rdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds Springer New York NY USA 1997
[3] V S Erturk and S Momani ldquoSolving systems of fractionaldifferential equations using differential transform methodrdquoJournal of Computational and AppliedMathematics vol 215 no1 pp 142ndash151 2008
[4] J Duan J An and M Xu ldquoSolution of system of fractionaldifferential equations by Adomian decomposition methodrdquoApplied Mathematics vol 22 no 1 pp 7ndash12 2007
[5] A S Mohamed and R A Mahmoud ldquoAn algorithm for thenumerical solution of systems of fractional differential equa-tionsrdquo International Journal of Computer Applications vol 65no 11 2013
[6] S Momani and R Qaralleh ldquoAn efficient method for solvingsystems of fractional integro-differential equationsrdquo Computersamp Mathematics with Applications vol 52 no 3-4 pp 459ndash4702006
[7] M Zurigat S Momani and A Alawneh ldquoHomotopy analysismethod for systems of fractional integro-differential equationsrdquoin Proceeding of the 4th International Workshop of AdvancedComputation for Engineering Applications pp 106ndash111 2008
[8] M Zurigat SMomani Z Odibat andA Alawneh ldquoThe homo-topy analysis method for handling systems of fractional differ-ential equationsrdquo Applied Mathematical Modelling vol 34 no1 pp 24ndash35 2010
[9] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007
[10] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 4 pp 451ndash459 2008
[11] Y Lin and Y Zhou ldquoSolving nonlinear pseudoparabolic equa-tions with nonlocal boundary conditions in reproducing kernelspacerdquo Numerical Algorithms vol 52 no 2 pp 173ndash186 2009
[12] B Ghazanfari and A G Ghazanfari ldquoSolving system of frac-tional differential equations by fractional complex transformmethodrdquoAsian Journal of Applied Sciences vol 5 no 6 pp 438ndash444 2012
[13] R K Saeed and H M Sadeq ldquoSolving a system of linear fred-holm fractional integro-differential equations using homotopyperturbation methodrdquo Australian Journal of Basic and AppliedSciences vol 4 no 4 pp 633ndash638 2010
[14] S Bushnaq S Momani and Y Zhon ldquoA reproducing KernelHilbert space method for solving integro-differential equationsof fractional orderrdquo Journal of Optimization Theory and Appli-cations vol 156 no 1 pp 96ndash105 2012
[15] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Hilbert Space Nova Science Publishers NewYork NY USA 2009
[16] J S Leszczynski An Introduction to Fractional MechanicsCzestochowa University of Technology Czestochowa Poland2011
[17] M Caputo ldquoLinear Models of Dissipation whose Q is almostFrequency Independent-IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967
[18] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006
[19] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000
[20] S G Samko A A Kilbas andO IMarichev Fractional Integraland Derivatives Theory and Applications Gordon and BreachScience Publishers Yverdon Switzerland 1993
[21] C-l Li and M-G Cui ldquoThe exact solution for solving a classnonlinear operator equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 143 no 2-3 pp393ndash399 2003
The exact solution for 1205721= 1205722= 1 is 119909(119905) = sinh 119905 119910(119905) =
cosh 119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 30 the graphs
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 2
Example 3 Consider the following nonlinear system offractional integrodifferential equations
1198631205721119909 (119905) = 1 minus
1199053
3minus1
211991010158402
(119905) +1
2int
119905
0
(1199092
(120591) + 1199102
(120591)) 119889120591
1198631205722119909 (119905) = minus1 + 119905
2
minus 119905119909 (119905) +1
4int
119905
0
(1199092
(120591) minus 1199102
(120591)) 119889120591
119909 (0) = 1 1199091015840
(0) = 2 119910 (0) = minus1
1199101015840
(0) = 0 1 lt 1205721 1205722le 2
(18)
The exact solution for 1205721= 1205722= 2 is 119909(119905) = 119905 + 1198903 119910(119905) =
119905 minus 119890119905After homogenizing the initial conditions and using this
method taking 119905119894= 119894119899 119894 = 1 2 119899 and 119899 = 20 the graphs
6 Abstract and Applied Analysis
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 3
4 Conclusion
In this paper we introduce a new algorithm for solving sys-tems of fractional integrodifferential equations The approxi-mate solution obtained by this method and its derivative areboth uniformly convergentThe obtained results demonstratethe reliability of the algorithm and its wider applicabilityto linear and nonlinear systems of fractional differentialequations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Podlubuy Fractional Differential Equations Academic PressNew York NY USA 1999
[2] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanics rdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds Springer New York NY USA 1997
[3] V S Erturk and S Momani ldquoSolving systems of fractionaldifferential equations using differential transform methodrdquoJournal of Computational and AppliedMathematics vol 215 no1 pp 142ndash151 2008
[4] J Duan J An and M Xu ldquoSolution of system of fractionaldifferential equations by Adomian decomposition methodrdquoApplied Mathematics vol 22 no 1 pp 7ndash12 2007
[5] A S Mohamed and R A Mahmoud ldquoAn algorithm for thenumerical solution of systems of fractional differential equa-tionsrdquo International Journal of Computer Applications vol 65no 11 2013
[6] S Momani and R Qaralleh ldquoAn efficient method for solvingsystems of fractional integro-differential equationsrdquo Computersamp Mathematics with Applications vol 52 no 3-4 pp 459ndash4702006
[7] M Zurigat S Momani and A Alawneh ldquoHomotopy analysismethod for systems of fractional integro-differential equationsrdquoin Proceeding of the 4th International Workshop of AdvancedComputation for Engineering Applications pp 106ndash111 2008
[8] M Zurigat SMomani Z Odibat andA Alawneh ldquoThe homo-topy analysis method for handling systems of fractional differ-ential equationsrdquo Applied Mathematical Modelling vol 34 no1 pp 24ndash35 2010
[9] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007
[10] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 4 pp 451ndash459 2008
[11] Y Lin and Y Zhou ldquoSolving nonlinear pseudoparabolic equa-tions with nonlocal boundary conditions in reproducing kernelspacerdquo Numerical Algorithms vol 52 no 2 pp 173ndash186 2009
[12] B Ghazanfari and A G Ghazanfari ldquoSolving system of frac-tional differential equations by fractional complex transformmethodrdquoAsian Journal of Applied Sciences vol 5 no 6 pp 438ndash444 2012
[13] R K Saeed and H M Sadeq ldquoSolving a system of linear fred-holm fractional integro-differential equations using homotopyperturbation methodrdquo Australian Journal of Basic and AppliedSciences vol 4 no 4 pp 633ndash638 2010
[14] S Bushnaq S Momani and Y Zhon ldquoA reproducing KernelHilbert space method for solving integro-differential equationsof fractional orderrdquo Journal of Optimization Theory and Appli-cations vol 156 no 1 pp 96ndash105 2012
[15] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Hilbert Space Nova Science Publishers NewYork NY USA 2009
[16] J S Leszczynski An Introduction to Fractional MechanicsCzestochowa University of Technology Czestochowa Poland2011
[17] M Caputo ldquoLinear Models of Dissipation whose Q is almostFrequency Independent-IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967
[18] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006
[19] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000
[20] S G Samko A A Kilbas andO IMarichev Fractional Integraland Derivatives Theory and Applications Gordon and BreachScience Publishers Yverdon Switzerland 1993
[21] C-l Li and M-G Cui ldquoThe exact solution for solving a classnonlinear operator equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 143 no 2-3 pp393ndash399 2003
of the approximate solutions for different values of 1205721and 120572
2
are plotted in Figure 3
4 Conclusion
In this paper we introduce a new algorithm for solving sys-tems of fractional integrodifferential equations The approxi-mate solution obtained by this method and its derivative areboth uniformly convergentThe obtained results demonstratethe reliability of the algorithm and its wider applicabilityto linear and nonlinear systems of fractional differentialequations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Podlubuy Fractional Differential Equations Academic PressNew York NY USA 1999
[2] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanics rdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds Springer New York NY USA 1997
[3] V S Erturk and S Momani ldquoSolving systems of fractionaldifferential equations using differential transform methodrdquoJournal of Computational and AppliedMathematics vol 215 no1 pp 142ndash151 2008
[4] J Duan J An and M Xu ldquoSolution of system of fractionaldifferential equations by Adomian decomposition methodrdquoApplied Mathematics vol 22 no 1 pp 7ndash12 2007
[5] A S Mohamed and R A Mahmoud ldquoAn algorithm for thenumerical solution of systems of fractional differential equa-tionsrdquo International Journal of Computer Applications vol 65no 11 2013
[6] S Momani and R Qaralleh ldquoAn efficient method for solvingsystems of fractional integro-differential equationsrdquo Computersamp Mathematics with Applications vol 52 no 3-4 pp 459ndash4702006
[7] M Zurigat S Momani and A Alawneh ldquoHomotopy analysismethod for systems of fractional integro-differential equationsrdquoin Proceeding of the 4th International Workshop of AdvancedComputation for Engineering Applications pp 106ndash111 2008
[8] M Zurigat SMomani Z Odibat andA Alawneh ldquoThe homo-topy analysis method for handling systems of fractional differ-ential equationsrdquo Applied Mathematical Modelling vol 34 no1 pp 24ndash35 2010
[9] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007
[10] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 4 pp 451ndash459 2008
[11] Y Lin and Y Zhou ldquoSolving nonlinear pseudoparabolic equa-tions with nonlocal boundary conditions in reproducing kernelspacerdquo Numerical Algorithms vol 52 no 2 pp 173ndash186 2009
[12] B Ghazanfari and A G Ghazanfari ldquoSolving system of frac-tional differential equations by fractional complex transformmethodrdquoAsian Journal of Applied Sciences vol 5 no 6 pp 438ndash444 2012
[13] R K Saeed and H M Sadeq ldquoSolving a system of linear fred-holm fractional integro-differential equations using homotopyperturbation methodrdquo Australian Journal of Basic and AppliedSciences vol 4 no 4 pp 633ndash638 2010
[14] S Bushnaq S Momani and Y Zhon ldquoA reproducing KernelHilbert space method for solving integro-differential equationsof fractional orderrdquo Journal of Optimization Theory and Appli-cations vol 156 no 1 pp 96ndash105 2012
[15] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Hilbert Space Nova Science Publishers NewYork NY USA 2009
[16] J S Leszczynski An Introduction to Fractional MechanicsCzestochowa University of Technology Czestochowa Poland2011
[17] M Caputo ldquoLinear Models of Dissipation whose Q is almostFrequency Independent-IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967
[18] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006
[19] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000
[20] S G Samko A A Kilbas andO IMarichev Fractional Integraland Derivatives Theory and Applications Gordon and BreachScience Publishers Yverdon Switzerland 1993
[21] C-l Li and M-G Cui ldquoThe exact solution for solving a classnonlinear operator equations in the reproducing kernel spacerdquoApplied Mathematics and Computation vol 143 no 2-3 pp393ndash399 2003