In this paper, variational homotopy perturbation iteration method (VHPIM) has been applied along with Caputo derivative to solve high-order fractional Volterra integro-differential equations (FVIDEs). The “VHPIM” is present in all two steps. In order to indicate the efficiency and simplicity of the proposed method, we have presented some examples. All of the numerical computations in this study have been done on a personal computer applying some programs written in Maple18.
Issue Section:
Research Papers
References
1.
Momani
, S.
, and Qaralleh
, R.
, 2006
, “An Efficient Method for Solving Systems of Fractional Integro-Differential Equations
,” Comput. Math. Appl.
, 52
(3–4
), pp. 459
–470
.10.1016/j.camwa.2006.02.0112.
Sweilam
, N. H.
, Khader
, M. M.
, and Al-Bar
, R. F.
, 2008
, “Homotopy Perturbation Method for Linear and Nonlinear System of Fractional Integro-Differential Equations
,” Int. J. Comput. Math. Numer. Simul.
, 1
(1
), pp. 73
–87
.3.
Das
, S.
, 2008
, Functional Fractional Calculus for System Identification and Controls
, Springer
, New York
.4.
Ganji
, Z. Z.
, Ganji
, D. D.
, and Esmaeilpour
, M.
, 2009
, “Study on Nonlinear Jeffery Hamel Flow by He's Semi-Analytical Methods and Comparison With Numerical Results
,” Comput. Math. Appl.
, 58
(11–12
), pp. 2107
–2116
.10.1016/j.camwa.2009.03.0445.
Momani
, S.
, and Noor
, M. A.
, 2006
, “Numerical Methods for Fourth-Order Fractional Integro-Differential Equations
,” Appl. Math. Comput.
, 182
(1
), pp. 754
–760
.10.1016/j.amc.2006.04.0416.
Sweilam
, N. H.
, Khader
, M. M.
, and Mahdy
, A. M. S.
, 2012
, “Numerical Studies for Fractional-Order Logistic Differential Equation With Two Different Delays
,” Appl. Math.
, 2012
, p. 764894
.7.
He
, J. H.
, 1998
, “Approximate Analytical Solution for Seepage Flow With Fractional Derivatives in Porous Media
,” Comput. Methods Appl. Mech. Eng.
, 167
(1–2
), pp. 57
–68
.10.1016/S0045-7825(98)00108-X8.
Sweilam
, N. H.
, Khader
, M. M.
, and Al-Bar
, R. F.
, 2007
, “Numerical Studies for a Multi-Order Fractional Differential Equation
,” Phys. Lett. A
, 371
(1–2
), pp. 26
–33
.10.1016/j.physleta.2007.06.0169.
Ganji
, D. D.
, and Rajabi
, A.
, 2006
, “Assessment of Homotopy-Perturbation and Perturbation Methods in Heat Radiation Equations
,” Int. Commun. Heat Mass Transfer
, 33
(3
), pp. 391
–400
.10.1016/j.icheatmasstransfer.2005.11.00110.
Ganji
, D. D.
, and Sadighi
, A.
, 2006
, “Application of He's Homotopy-Perturbation Method to Nonlinear Coupled Systems of Reaction-Diffusion Equations
,” Int. J. Nonlinear Sci. Numer. Simul.
, 7
(4
), pp. 411
–418
.10.1515/IJNSNS.2006.7.4.41111.
Golbabai
, A.
, and Javidi
, M.
, 2007
, “A Third-Order Newton Type Method for Nonlinear Equations Based on Modified Homotopy Perturbation Method
,” Math. Comput.
, 191
(1
), pp. 199
–205
.12.
Hashim
, I.
, 2006
, “Adomian Decomposition Method for Solving BVPs for Fourth-Order Integro-Differential Equations
,” J. Comput. Appl. Math.
, 193
(2
), pp. 658
–664
.10.1016/j.cam.2005.05.03413.
Hashim
, I.
, Abdulaziz
, O.
, and Momani
, S.
, 2009
, “Homotopy Analysis Method for Fractional IVPs
,” Commun. Nonlinear Sci. Numer. Simul.
, 14
(3
), pp. 674
–684
.10.1016/j.cnsns.2007.09.01414.
Khader
, M. M.
, 2011
, “On the Numerical Solutions for the Fractional Diffusion Equation
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
(6
), pp. 2535
–2542
.10.1016/j.cnsns.2010.09.00715.
Khader
, M. M.
, 2013
, “Numerical Treatment for Solving Fractional Riccati Differential Equation
,” J. Egypt. Math. Soc.
, 21
(1
), pp. 32
–37
.10.1016/j.joems.2012.09.00516.
Khader
, M. M.
, 2013
, “Numerical Treatment for Solving the Perturbed Fractional PDEs Using Hybrid Techniques
,” J. Comput. Phys.
, 250
, pp. 565
–573
.10.1016/j.jcp.2013.05.03217.
He
, J. H.
, 1999
, “Homotopy Perturbation Technique
,” Comput. Methods Appl. Mech. Eng.
, 178
(3–4
), pp. 257
–262
.10.1016/S0045-7825(99)00018-318.
He
, J. H.
, 2000
, “A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems
,” Int. J. Non-Linear Mech.
, 35
(1
), pp. 37
–43
.10.1016/S0020-7462(98)00085-719.
Samko
, S.
, Kilbas
, A.
, and Marichev
, O.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
, Gordon and Breach
, London
.20.
Gutiérrez
, R. E.
, Rosário
, J. M.
, and Tenreiro Machado
, J. A.
, 2010
, “Fractional Order Calculus: Basic Concepts and Engineering Applications
,” Math. Probl. Eng.
, 2010
, p. 375858
.10.1155/2010/37585821.
Valerio
, D.
, Trujillo
, J. J.
, Rivero
, M.
, Tenreiro Machado
, J. A.
, and Baleanu
, D.
, 2013
, “Fractional Calculus: A Survey of Useful Formulas
,” Eur. Phys. J.: Spec. Top.
, 222
(8
), pp. 1827
–1846
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