The Application Of Lie Symmetry Analysis In Solving Several Kinds Of Partial Differential Equations

Since the 19 th century,the study of the solutions of nonlinear partial differential equations has become a hot spot with the nonlinear partial differential equations being used to the objective world widely.However,it is extremely challenging that how to obtain the solutions of nonlinear partial differential equations.Although many methods have been proposed for obtaining the solutions of nonlinear partial differential equations,there is no unified and universal method for partial differential equations.So it is necessary and important to find some more effective and feasible methods.Based on this purpose,this paper researchs three kinds of nonlinear partial differential equations with Lie symmetry methods: ill-posed Boussinesq equation,two-component CamassaHolm equation and the fifth-order time-fractional KDV equation.And we have obtained the Lie symmetry,vector field,similarity reduction of these equations according to the basic thought of the Lie symmetry method.Furthermore,the reduced equations corresponding to these equations are obtained.At the same time,according to the relevant theory of power series,the power series solutions of these equations are obtained.The convergence of the resulting power series solution is proved.The main contents of this paper are as follows:The first chapter is the preliminariy.It reviews the research background and several methods for solving the partial differential equation briefly.Then,it introduces the research background of Lie symmetry analysis method and elaborates the basic idea of Lie symmetry method.The second chapter is propaedeutic.The relevant definitions about Lie symmetry analysis method and the fractional partial differential equation are given.The third chapter contains the solution of ill-posed Boussinesq equation by Lie symmetry analysis.This chapter gives the specific expression and physical meaning of ill-posed Boussinesq equation,and it transforms the ill-posed Boussinesq equation into ordinary differential equation by Lie symmetry analysis.The corresponding power series solution of the equation is obtained and the convergence of the power series solution is also proved.The fourth chapter uses symmetry analysis method to solve the two-component Camassa-Holm equation.It also gives the concrete form of this equation and the physical significance.At the same time,Lie symmetry method is applied to obtain the several different reduced equation of two-component Camassa-Holm equation.And reduced process of the power series solution of the equation is given and its convergence is proved.The fifth chapter obtains the solution of the fifth-order time-fractional KDV equation by Lie symmetry analysis.The fifth-order time-fractional KDV equation is analyzed by combining of the preliminary knowledge and the basic theory of the fractional partial differential equation.Then,the fifth-order time-fractional KDV equation is transformed into the corresponding fractional order ordinary differential equation.The details of the conversion process is also given.Finally,it also obtains the power series solution of the corresponding fractional order differential equation and proves the convergence of the power series solution.The sixth chapter is summary and outlook.Main research content of this article is summarized and it puts forward some problems to solve,the following research direction is also prospected.