%0 Journal Article
%T Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations
%A Lei Zhang
%A Lisha Wang
%A Xiaohua Ding
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/597926
%X We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes. 1. Introduction During the last few decades, nonlinear diffusion equation (1) has played an important role in nonlinear physics. Recently, it also began to become important in various other fields of science, for example, biology, chemistry, and economics [1¨C3]. When , (1) is reduced to the famous Burgers equation (2) This equation is the simplest equation combining both nonlinear propagation effects and diffusive effects [3]. It has been used in many fields especially for describing wave processes in acoustics and hydrodynamics [2]. Researchers have devoted a lot of efforts to studying the solutions of this equation [1¨C6]. A. van Niekerk and F. D. van Niekerk [4] applied Galerkin methods to the nonlinear Burgers equation and obtained implicit and explicit algorithms using different higher order rational basis functions. Hon and Mao [5] applied the multiquadric as a spatial approximation scheme for solving the nonlinear Burgers equation. Biazar and Aminikhah [6] considered the variational iteration method to solve nonlinear Burgers equation. If we take , (1) becomes the Burgers-Fisher equation (3) Burgers-Fisher equation is very important in fluid dynamic model. There have been extensive studies and applications of this model. A nonstandard finite difference scheme for the Burgers-Fisher equation was given by Mickens and Gumel [7]. In [8], Kaya and El-Sayed constructed a numerical simulation and explicit solutions of the generalized Burgers-Fisher equation. Ismail et al. [9] obtained the approximate solutions for the Burgers-Huxley and Burgers-Fisher equations by using the Adomian decomposition method. Wazwaz [10] presented the tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Javidi and Golbabai [11, 12] studied spectral collocation method and spectral domain decomposition method for the solution of the generalized Burgers-Fisher equation. Numerical solution of Burgers-Fisher equation is presented based on the cubic B-spline quasi-interpolation by Zhu and Kang [13]. Kocacoban et al. [14] solved Burgers-Fisher equation by using a different numerical approach that shows rather rapid convergence than other
%U http://www.hindawi.com/journals/jam/2014/597926/