Two Difference Schemes For The Initial Boundary Value Problem Of Degasperis-Procesi Equation

In recent years,the theory of partial differential equations has been widely used.Many complex models have been studied theoretically by establishing partial differential equations.In reality,the exact solutions of most partial differential equations are difficult to be obtained.At this point,we can only solve the differential equation by approximate methods.With the rapid development of artificial intelligence,the Internet of Things and cloud computing,the numerical solution of partial differential equations has also achieved unprecedented development.Difference method is one of the common methods to solve the definite solution of partial differential equation.By constructing a feasible difference scheme to find approximate solution,the numerical solution of the equation is analyzed theoretically.In this paper,we mainly study the finite difference scheme for the initial boundary value problem of Degasperis-Procesi equation,and analyze the conservation,existence and uniqueness,convergence and stability of the solution of the difference scheme to prove the feasibility of the difference scheme.The specific content of the article is as follows:In the first chapter,the research background and status of finite difference scheme and Degasperis-Procesi equation are described.In the second chapter,the establishment method of finite difference scheme and its related definition and lemma are given,and the symbol representation suitable for this paper is given.In Chapter 3 and Chapter 4,two-layer finite difference schemes and three-layer finite difference schemes are established for the initial boundary value problem of Degasperis-Procesi equation.By looking for the appropriate conserved quantity,the conservation of the solution of the difference scheme is proved,and Browder's theorem is used to prove the existence of the solution of the difference scheme.The uniqueness,convergence and stability of the solution are analyzed,and the difference scheme is proved to be second-order convergence.Finally,the corresponding iteration scheme is given,and a numerical example is given to verify the feasibility of the schemes.In the fifth chapter,the thesis is summarized.