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.

References

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.2006.02.011
2.
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.2009.03.044
5.
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.2006.04.041
6.
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-X
8.
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.2007.06.016
9.
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.2005.11.001
10.
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.2006.7.4.411
11.
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.2005.05.034
13.
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.2007.09.014
14.
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.2010.09.007
15.
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.2012.09.005
16.
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.2013.05.032
17.
He
,
J. H.
,
1999
, “
Homotopy Perturbation Technique
,”
Comput. Methods Appl. Mech. Eng.
,
178
(
3–4
), pp.
257
262
.10.1016/S0045-7825(99)00018-3
18.
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)00085-7
19.
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
.10.1155/2010/375858
21.
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
.
You do not currently have access to this content.