Study On The Solution Of Exact Solution Of Nonlinear Partial Differential Equation

In the second half of the twentieth century, in view of the nonlinear partial differ-ential equations can be more accurate description of the widespread phenomenon of the objective world, so study of nonlinear partial differential equations has gradually become hot. When people are continuously exploring and solving the nonlinear problems,the un-derstanding of nature and society is more and more profound. Up to now, the exploration of nonlinear partial differential equation has established a relatively complete theoretical system.In the research field of nonlinear partial differential equations, the construction of ex-act solutions is one important branch. Since the past few decades, the development of the exact solutions has new breakthrough, scientists have proposed many effective methods to construct the exact solutions of the equations. What is particularly worth mentioning is, Chinese scientists have made outstanding achievements in this area, is in the leading position in the world, such as the homogeneous balance method founded by professor Wang Mingliang, this method can be used as the basis of many other methods; Multi linear variable separation approach established by professor Lou Senyue; The extended tanh method proposed by professor Fan Engui; Jacobi elliptic expansion function created by professor Liu and this thesis will introduce the double function method and G’/G-expansion method and so on. These pioneering works have greatly enriched the methods for solving exact solutions of nonlinear partial differential equations, and these methods can be applied to a large type of nonlinear equations, which are of high academic value.The symbol system plays an vital role in getting traveling wave solutions of nonlinear partial differential equation. Combined with the symbolic computation software Maple, this thesis will give the 3D graphs of each group of exact travelling wave solutions. It enables us to analyze the shape, properties, meaning of the solutions more easily.In this thesis, we mainly study from the following several aspects:In section one, we introduce the background and significance of the research.In section two, firstly we discuss the soliton generation and excitation, illustrate the influence of soliton on the nonlinear system of partial differential equations. Then, we will introduce several methods of solving nonlinear partial differential equations. At last, give the generalized conditional symmetry and conservation law of the nonlinear partial differential equations, which plays an important role in the integration, the linearization of nonlinear partial differential equations.Section three introduces the basic idea of the double function method and its specific solving steps, and then the new travelling wave solutions of partial differential equation of high order Kawahara are obtained by using this method, that enrich the exact solutions of nonlinear partial differential equations. With Maple, plot the 3D graph of each solution to verify the correctness of them.Section four first combines the Riemann-Liouville fractional derivative and G’/G-expansion method to find new exact solutions of the high-order fractional Kawahara equa-tion,then analyzes the meaning of the solutions, before it, some definitions and properties of this two kinds of methods given out. Finally compared with G’/G-expansion method and other methods to get their advantages and disadvantages.Section five is a summary of this thesis and the prospect of solving nonlinear partial differential equations.