%0 Journal Article
%T Discrete Symmetries Analysis and Exact Solutions of the Inviscid Burgers Equation
%A Hongwei Yang
%A Yunlong Shi
%A Baoshu Yin
%A Huanhe Dong
%J Discrete Dynamics in Nature and Society
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/908975
%X We discuss the Lie point symmetries and discrete symmetries of the inviscid burgers equation. By employing the Lie group method of infinitesimal transformations, symmetry reductions and similarity solutions of the governing equation are given. Based on discrete symmetries analysis, two groups of discrete symmetries are obtained, which lead to new exact solutions of the inviscid Burgers equation. 1. Introduction Burgers equation is one of the basic partial differential equations of fluid mechanics. It occurs in various fields of applied mathematics, such as modeling of gas dynamics and traffic flow. For a given velocity and viscosity coefficient , the general form of Burgers equation is: where is a smooth function of . If , Burgers equation reduces to the inviscid Burgers equation: which is a prototype for equations for which the solution can develop discontinuities (shock waves). There are many methods to solve (1.2). In [1], the authors discussed the matrix exponential representations of solutions to similar equation of (1.2). Here we can use the method of Lie symmetries and discrete symmetries analysis to solve (1.2). The classical Lie symmetries of the partial differential equations (PDEs) which can be obtained through the Lie group method of infinitesimal transformations were originally developed by Lie [2]. We can use the basic prolongation method and the infinitesimal criterion of invariance to find some particular Lie point symmetries group of the nonlinear partial differential equations. The Lie groups of transformations admitted by a given system of differential equations can be used (1) to lower the order or eventually reduce the equation to quadrature, in the case of ordinary differential equations; (2) to determine particular solutions, called invariant solutions, or generate new solutions, once a special solution is known, in the case of ordinary differential equations or PDEs. In the past decades, much attention has been paid to the symmetry method and a series of achievements have been obtained [3¨C9]. Particularly, In [9], a five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schr£żdinger equation. But it seems that very few research on discrete symmetries is available up to now. In fact, discrete symmetries also play an important role in solving PDEs. For instance, to understand how a system changes its stability, to simplify the numerical computation of solutions of PDEs and to create new exact solutions from known solutions. Discrete symmetries are usually easy to guess but
%U http://www.hindawi.com/journals/ddns/2012/908975/