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Homotopy analysis method for space‐time fractional differential equations

Xindong Zhang (College of Mathematics and System Sciences, Xinjiang University, Urumqi, People's Republic of China)
Leilei Wei (Center for Computational Geosciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, People's Republic of China)
Bo Tang (Center for Computational Geosciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, People's Republic of China)
Yinnian He (Center for Computational Geosciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, People's Republic of China)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 2 August 2013

175

Abstract

Purpose

In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial conditions.

Design/methodology/approach

The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions.

Findings

The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior.

Originality/value

The idea can be used to find the numerical solutions of other fractional differential equations.

Keywords

Citation

Zhang, X., Wei, L., Tang, B. and He, Y. (2013), "Homotopy analysis method for space‐time fractional differential equations", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 23 No. 6, pp. 1063-1075. https://doi.org/10.1108/HFF-09-2011-0181

Publisher

:

Emerald Group Publishing Limited

Copyright © 2013, Emerald Group Publishing Limited

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