A Nonlinear Perturbation Problem For Wave Equation

The wave equation is one of the most comprehensive scientific theses. Most physical problems can be described as nonlinear wave equation, and its nonlinear term only depend on the derivatives of some dependent variables and. small pararneterε, just as Rayleigh wave equation. Thus, it becomes one of the most important contents of the nonlinear science to search for the exact solution of nonlinear partial differential equation in mathematical physics, specially, the nonlinear perturbation problem for wave equation. Generally, it is very difficult to obtain the exact solution of nonlinear partial differential equation, especially with perturbation. Furthermore, there is a lack of the research on this area presently; hence, it is very necessary to solve the solution for this kind of nonlinear equations.This paper contains three chapters, chapter 1 is introduction, firstly, the background and importance of nonlinear perturbation problem for wave equation is introduced; and then, the researches in this area is presented; finally, the methods used in chapter 2 are presented, namely, the method of multiple scales and energy method. Chapter 2, the initial value problem for the Rayleigh wave equation is considered: and a nonlinear perturbation problem for wave equation is studied. First of all, the global existence and uniqueness of the solution is obtained by using the integral equation and Banach's fixed point theorem; and then, by virtue of the energy method, integral equation and Banach's fixed point theorem, a priori uniform estimate for the solution is obtained; at last, the difference between the true solution and the approximate solution is estimated.Chapter 3 will discuss P.C. Iterative Form of Newton Secant Method as follows: and proved its order of convergence to be 2.618.