Lie Group Analysis And Power Series Method For Dealing With Nonlinear Partial Differential Equations

In this paper,using /G'G-expansion method and the classical Lie group method,we mainly study the following three groups of nonlinear partial differential equations: the extended Burgers' equation,the nonlinear LC circuit equation,the high-order nonlinear wave equation.Firstly,we obtain a lot of symmetries of the equation.Secondly,using the symmetries,we find some classical reductions of the equations.Finally,many kinds of new exact solutions are derived by solving the reduced equations.In Chapter 1,using /G'G-expansion method for dealing with extended Burgers' equation.The existence of the possible /G'G-expansion solutions to the extended Burgers' equations are discussed in detail,and all of the /G'G solutions to the equations are obtained.In Chapter 2,using the classical Lie group analysis theory,we find the classical Lie symmetry of nonlinear LC circuit equation equation.At the same time,a great many of exact solutions are derived by reduction equation.Including the solutions,the power series solutions and so on.In Chapter 3,using the classical Lie group analysis theory,we find the classical Lie symmetry of high-order nonlinear wave equation.At the same time,an optimal system of the equation is obtained by Lie symmetry analysis method and the exact solutions to the equation are provided by the power series method.Finally,a lot of conservation laws of equation are presented by using symmetries of the equation.To sum up,the main thought of this paper is using Lie symmetry analysis theory and method to solve the nonlinear partial differential equations.Using Lie symmetry analysis theory can effectively solve the nonlinear partial differential equations and we obtain new exact solutions of nonlinear partial differential equations.Finally,the conservation laws ofnonlinear partial differential equations are obtained by using the vector fields of equation and a lot of theorems.