A computational technique for the solution of high-order fractional Volterra integro-differential equations

Bahram Agheli; Rahmat Darzi

doi:10.26637/MJM0602/0017

Abstract

The optimal q-homotopy analysis method has been employed in order to solve high-order Volterra integrodifferential equations featuring time-fractional derivative. Then, in order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica.

Keywords:

Nonlinear fractional integro-differential equation, Optimal q-homotopy analysis method, Caputo derivative

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Bahram Agheli Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.
Rahmat Darzi Department of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran.

Pages: 402-407
Date Published: 01-04-2018
Vol. 6 No. 02 (2018): Malaya Journal of Matematik (MJM)

Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear problems (Doctoral dissertation, Ph. D. Thesis, Shanghai Jiao Tong University).

El-Tawil, M. A. and Huseen, S.N. (2012). The qHomotopy Analysis Method (q-HAM), International Journal of Applied mathematics and mechanics, 8 (15): 51-75.

Huseen S. N., Grace S. R. and El-Tawil M. A. (2013). The Optimal q-Homotopy Analysis Method (Oq-HAM), International Journal of Computers & Technology, Vol 11, No. 8.

Kilbas, A. A., Srivastava, H. M., and Trujillo, J.J., (2006). Theory and application of fractional differential equations, Elsevier B.V, Netherlands.

Miller, K. S., and Ross, B. (1993). An introduction to the fractional calculus and fractional differential equation, John Wiley and Sons, New York.

He, J.H. (1998) Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. and Eng., 167 (1-2) 57 68 .

Sweilam, N.H., Khader, M.M., Al-Bar, R.F. (2007) Numerical studies for a multi-order fractional differential equation, Phys. Lett. A, 371 26-33.

Ganji,D.D., Rajabi, A. (2006) Assessment of homotopyperturbation and perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transfer, 33 (3) 391400.

Ganji, D.D., 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) 411-418.

Golbabai, A., Javidi, M. (2007) A third-order Newton type method for nonlinear equations based on modified homotopy perturbation method, Mathematics and Computation, 191 199-205.

Hashim, I. (2006) Adomian decomposition method for solving BVPs for fourth-order integro-differential equations, J. Comput. Appl. Math., 193 658-664.

Hashim, I., Abdulaziz, O., Momani, S. (2009) Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14 674-684.

Khader, M.M. (2011) On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16 2535-2542.

Khader, M.M. (2013) Numerical treatment for solving fractional Riccati differential equation, J. Egyptian Math. Soc., 21 32-37.

Khader, M.M. (2013) Numerical treatment for solving the perturbed fractional PDEs using hybrid techniques, J. Comput. Phys., 250 565-573.

Yabushita, K., Yamashita, M., & Tsuboi, K. (2007). An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. Jour- nal of Physics A: Mathematical and theoretical, 40(29), 8403.

El-Tawil, M. A., & Huseen, S. N. (2013). On convergence of the q-homotopy analysis method. International Journal of Contemporary Mathematical Sciences, 8 (10), 481497.

Neamaty, A., Agheli, B., & Darzi, R. (2015). Numerical Solution of High-Order Fractional Volterra IntegroDifferential Equations by Variational Homotopy Perturbation Iteration Method. Journal of Computational and Nonlinear Dynamics, 10(6), 061023 .