Homotopy analysis method for space‐ and time‐fractional KdV equation

Abstract

Purpose

The purpose of this paper is to present numerical solutions for the space‐ and time‐fractional Korteweg‐de Vries (KdV) equation using homotopy analysis method (HAM). The space and time‐fractional derivatives are described in the Caputo sense. The paper witnesses the extension of HAM for fractional KdV equations.

Design/methodology/approach

This paper presents numerical solutions for the space‐ and time‐fractional KdV equation using HAM. The space and time‐fractional derivatives are described in the Caputo sense.

Findings

In this paper, the application of homotopy analysis method was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The homotopy analysis method was clearly a very efficient and powerful technique in finding the solutions of the proposed equations.

Originality/value

In this paper, the application of HAM was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The HAM was clearly very efficient and powerful technique in finding the solutions of the proposed equations. The obtained results demonstrate the reliability of the algorithm and its wider applicability to fractional nonlinear evolution equations. Finally, the recent appearance of nonlinear fractional differential equations as models in some fields such as the thermal diffusion in fractal media makes it necessary to investigate the method of solutions for such equations and the authors hope that this paper is a step in this direction.

Citation

Tauseef Mohyud‐Din, S., Yıldırım, A. and Yülüklü, E. (2012), "Homotopy analysis method for space‐ and time‐fractional KdV equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 22 No. 7, pp. 928-941. https://doi.org/10.1108/09615531211255798