Modified simple equation method for nonlinear evolution equations Anwar Ja’afar Mohamad Jawad a , Marko D. Petkovic ´ b , Anjan Biswas c, * a Department of Electrical Engineering, University of Technology, PO Box 35010, Baghdad 00964, Iraq b Faculty of Science and Mathematics, Department of Mathematics and Computer Science, University of Niš, Višegradska 33, 18000 Niš, Serbia c Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA article info Keywords: Solitons Exact solutions abstract This paper reflects the implementation of a reliable technique which is called modified simple equation method (MSEM) for solving evolution equations. The proposed algorithm has been successfully tested on two very important evolution equations namely Fitzhugh– Nagumo equation and Sharma–Tasso–Olver equation. Numerical results are very encouraging. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction This paper outlines the implementation of modified simple equation method (MSEM) for solving evolution equations [1–12] which are very important in applied sciences. The MSEM is implemented on Fitzhugh–Nagumo equation and Sharma–Tasso–Olver equation. It is to be highlighted that Fitzhugh–Nagumo equation [5] implemented in the nerve prop- agation in biology, genetics, biology, and heat and mass transfer. The Sharma–Tasso–Olver equations appear in quantum field theory, relativistic physics, dispersive wave-phenomena, plasma physics, nonlinear optics and applied and physical sciences [2]. 2. Modified simple equation method (MSEM) The modified simple equation method is applied to find out an exact solution of a nonlinear ordinary differential equation Pðu; u 0 ; u 00 ; ...Þ¼ 0; ð1Þ where u = u(z) is an unknown function, P is a polynomial of the variable u and its derivatives. To solve (1) we expand its solutions u(z) in a finite series uðzÞ¼ X N k¼0 A k w z w  k ; A k ¼ const; A N – 0: ð2Þ The basic idea of the simple equation method is the assumption are not only some special functions, but they are the func- tions that satisfy some ordinary differential equations. These ordinary differential equations are referred to as the simplest equations. Two main features characterize the simplest equation: first, this is the equation of a lesser order than (1); second, the general solution of this equation is known. This means that the exact solutions u(z) of (1) can be presented by a finite series (2) in the general solution of the simplest equation. The main steps of our algorithm are: 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.030 * Corresponding author. E-mail address: biswas.anjan@gmail.com (A. Biswas). Applied Mathematics and Computation 217 (2010) 869–877 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc