A Better Approximation to the Solution of Burger-Fisher Equation D. KOCACOBAN, A.B. KOC, A. KURNAZ , and Y. KESKİN Abstract—The Burger-Fisher equations occur in various areas of applied sciences and physical applications, such as modeling of gas dynamics, financial mathematics and fluid mechanics. In this paper, this equation has been solved by using a different numerical approach that shows rather rapid convergence than other methods. Illustrative examples suggest that it is a powerful series approach to find numerical solutions of Burger-Fisher equations. Index Terms—Reduced differential transform method, Variational iteration method, Burger-Fisher Equation. I. INTRODUCTION HE Burger-Fisher equation has important applications in various fields of financial mathematics, gas dynamic, traffic flow, applied mathematics and physics applications[8-16]. This equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffussion transport[7]. The Burger-Fisher equation uncovers Johannes Martinus Burgers (1895-1981) and Ronald Aylmer Fisher (1890- 1962). In this paper, our aim is to solve the Burger-Fisher equation using Reduced Differential Transformation Method (RDTM)[1]-[5] and to compare the results with those of the exact solution. D. Kocacoban is with Department of Mathematics, Selcuk University, Konya, 42075 TURKEY (corresponding author to provide phone: +90- 554-4650609; e-mail: durdanekocacoban@gmail.com). A. B. KOC is with Department of Mathematics, Selcuk University, Konya, 42075 TURKEY (corresponding author to provide phone: +90- 533-5109082; e-mail: aysebetulkoc@yahoo.com). A. KURNAZ is with Department of Mathematics, Selcuk University, Konya, 42075 TURKEY (corresponding author to provide phone: +90- 505-8176657; e-mail: akurnaz@selcuk.edu.tr). Y. KESKİN is with Department of Mathematics, Selcuk University, Konya, 42075 TURKEY (corresponding author to provide phone: +90- 505-5760378; e-mail: yildiraykeskin@yahoo.com). The standart Burger-Fisher equation[6] can be written as ( 1) 0, t xx x u u uu uu          0 1, 0, x t    (1.1)      1 1 2 tanh 2 1 2 1 ) 0 , (                    x x u (1.2)               1 2 2 1 1 1 2 tanh 2 1 2 1 ) , (                                        t x t x u (1.3) where,    , , are non-zero parameters and k k k x x u    ) ( . The proposed method in the solution process of this equation has been successfully applied to solve many types of linear and nonlinear equation as Kawahara, Gas dynamics, Nonlinear dispersive K   n m, , Generalized Hirota-Satsuma coupled KdV and Coupled Modified KdV equations ([1]-[5]). II. ANALYSIS OF THE METHOD The basic definition of RDTM and that of its inverse can be given respectively as follow [3]: Definition 2.1. If two dimensional function   , uxt is analytic over a specified interval of time t and spatial dimension x , then we define   1 () , ! 0 k U x uxt k k k t t             (2.1) where the t-dimensional spectrum function   U x k is called the transformed function of u. Throughout this paper, the lowercase   , uxt represents the original function while the uppercase   U x k stands for the transformed function with respect to time variable t. T Proceedings of the World Congress on Engineering 2011 Vol I WCE 2011, July 6 - 8, 2011, London, U.K. ISBN: 978-988-18210-6-5 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2011