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

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solvingnonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solvingthe generalized Fitzhugh-Nagumo equation with time-dependentcoefficients. As a result, many new and more general exacttravelling wave solutions are obtained including periodic functionsolutions, soliton-like solutions, and trigonometric functionsolutions. 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. Theresults demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method andTanh method applied by Triki and Wazwaz (2013). Recently, this methodis developed for searching exact travelling wave solutions ofnonlinear partial differential equations. Abundant exacttravelling wave solutions including solitons, kink, and periodic andrational solutions have been found. These solutions might play animportant role in engineering fields. It is shown that thismethod, with the help of symbolic computation, provides astraightforward and powerful mathematical tool for solving thenonlinear physics