Table of Contents
International Journal of Engineering Mathematics
Volume 2015, Article ID 107978, 35 pages
http://dx.doi.org/10.1155/2015/107978
Research Article

New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

Department of Applied Mathematics, Faculty of Mathematics Science, University of Tabriz, Tabriz 51569 84687, Iran

Received 30 May 2015; Revised 10 September 2015; Accepted 10 September 2015

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2015 Jalil Manafian and Mehrdad Lakestani. 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.

Abstract

An improvement of the expansion methods, namely, the improved -expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new -expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.