New numerical technique for solving the fractional Huxley equation

Abstract

Purpose

The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation with time-fractional derivative.

Design/methodology/approach

Objectives achieved the main methods: the fractional derivative of f (x) in the Caputo sense is first stated. Second, the time-fractional Huxley equation is written in a differential operator form where the differential operator is in Caputo sense. After acting on both sides by the inverse operator of the fractional differential operator in Caputo sense, the Adomian's decomposition is then used to get the power series solution of the resulted time-fractional Huxley equation. Also, a second objective is achieved by applying the variational iteration method to get approximate solutions for the time-fractional Huxley equation.

Findings

There are some important findings to state and summarize here. First, the variational iteration method and the decomposition method provide the solutions in terms of convergent series with easily computable components for this considered problem. Second, it seems that the approximate solution of time-fractional Huxley equation using the decomposition method converges faster than the approximate solution using the variational iteration method. Third, the variational iteration method handles nonlinear equations without any need for the so-called Adomian polynomials. However, Adomian decomposition method provides the components of the exact solution, where these components should follow the summation given in Equation (21).

Originality/value

This paper presents new materials in terms of employing the variational iteration and the Adomian decomposition methods to solve the problem under consideration. It is expected that the results will give some insightful conclusions for the used techniques to handle similar fractional differential equations. This emphasizes the fact that the two methods are applicable to a broad class of nonlinear problems in fractional differential equations.

Citation

El-Sayed El-Danaf, T., Ali Zaki, M. and Moenaaem, W. (2014), "New numerical technique for solving the fractional Huxley equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24 No. 8, pp. 1736-1754. https://doi.org/10.1108/HFF-07-2013-0216