The Research On Methods For Solving Nonlinear Partial Differential Equations

Nonlinear dynamics is one of the important branches in nonlinear science, whereas study on how to solve nonlinear partial differential equations is one of the main contents in nonlinear dynamics. As a leading subject and hot interest in nonlinear science, study on the solution method of nonlinear partial differential equations has become more and more challenging. At present, although a number of methods are proposed and developed to look for the exact solutions of nonlinear partial differential equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear partial differential equations directly. As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear partial differential equations. In this thesis, based on a complete summary and examination of the main methods for solving nonlinear partial differential equations, a few new approaches are proposed to seek the exact solutions of nonlinear partial differential equations. By making use of these approaches proposed by us, a variety of exact solutions to many physically significant nonlinear partial differential equations are easily presented. Among them, some are in very good agreement with those obtained in some literatures; others are new ones which can not be found in the existing literatures to the best of our knowledge. The studies are of more profoundly theoretical significance and important application value.This dissertation consists of six chapters. In chapter one, the historical background of nonlinear partial differential equations is looked back on, the main methods for solving of nonlinear partial differential equations are summarized, and the primary contents of this dissertation are reported as well. In chapter two, several important nonlinear partial differential equations in mechanics are derived. In chapter three, a new approach called the transformation-trial function method is proposed to solve nonlinear partial differential equations, and many solutions including new ones to varieties of nonlinear partial differential equations are easily found by means of this approach. In chapter four, another new approach named as the direct method is put forward to seek the exact solutions of the Burgers equation and KdV equation together with KdV-Burgers equation, some of which are novel solutions which can not be obtained by the transformation-trial function method in chapter three. In chapter five, a third new approach called the superposition method is proposed to construct...