Communications in Numerical Analysis

Volume 2015, No. 2 (2015), Pages 115-124

Article ID cna-00239, 10 Pages

doi: 10.5899/2015/cna-00239

Research Article

Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

Yasser. S. Hamed1,3 *, Mohamed S. Mohamed2,3, E. R. El-Zahar4,5

1Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt

2Department of Mathematics, Faculty of Science, Al Azhar University, Cairo 11884, Egypt

3Department of Mathematics and Statistics, Faculty of Science, Taif University,21974, Kingdom of Saudi Arabia

4Department of Mathematics, College of Sciences and Humanities, Salman Bin Abdulaziz University, P.O. Box 83, Alkharj 11942, KSA

5Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menofia University, Egypt

* Corresponding author. Email address:

Received: 31 March 2015; Revised: 28 June 2015; Accepted: 07 July 2015

Copyright © 2015 Yasser. S. Hamed, Mohamed S. Mohamed and E. R. El-Zahar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this article, fractional complex transform with optimal homotopy analysis method (OHAM) is used to obtain numerical and analytical solutions for the nonlinear time-space fractional Fornberg-Whitham. Fractional complex transform is proposed to convert time-space fractional Fornberg-Whitham equation to the nonlinear ordinary differential equations and then applied OHAM to the new obtained equations. This optimal approach has general meaning and can be used to get fast convergent series solutions of the different type of nonlinear fractional differential equation. The results reveal that the method is very effective, simple and the OHAM contains a certain auxiliary parameter $h$ which provides us with a simple way to adjust and control the convergence region of convergence of the series solution.

Keywords: Homotopy analysis method; Optimal value; Fractional complex transform; Time-space fractional Fornberg-Whitham equation.


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