Analysis & Application To Some Methods For Solving Nonlinear Partial Differential Equations

With the developing of science and technology, the research for nonlinear problem has run through information science, life science, space science, geographic science, environmental science and many other scientific areas, because this research has not only great theoretical significance, but also practical meaning for the social progress and development.Nonlinear problem is very complex. To study nonlinear phenomenon, first at all, appropriate mathematical models must be established on the basis of actual problem, and many mathematical models in physical science, engineering technology and many applied scientific areas come down to differential equation, so studying the property and solving methods of differential equations has become one of the hot issues for the current scientific research, especially for nonlinear scientific research.After decades of development, many approaches have been found to solve nonlinear partial differential equations. In this dissertation, With the help of computer symbols system Mathematica, some methods of studying and solving nonlinear partial differential equations christened the first integral method, similarity transformation method and Painleve analysis or Painleve test technique are itemized and used, and the following three aspect tasks are completed: Firstly, by using the first integral method, we obtain a series of new exact solutions of the Fitzhugh-Nagumo equation. We also solve the Fisher equation with the same method, and four groups of exact solutions for this equation are obtained.Secondly, similarity reductions and similarity solutions of a nonlinear dispersive-dissipative equation are discussed by three different similarity transformation methods respectively, and several different types of similarity reductions and a new similarity solution of this equation are given.Thirdly, applying the WTC method to a nonlinear dispersive- dissipative equation, we find that this equation can not possess Painleve property, and therefore it is not to be completely integrable.The dissertation includes four chapters. In chapterâ… , some background knowledge and several common solving methods of nonlinear partial differential equations are introduced, which include inverse scattering method, Hirota bilinear method, Tanh function method, a unified algebra method based on the symbols computation and homogeneous balance method, and so on.In chapterâ…¡, the basic principles and major steps of the first integral method are expatiated, and then two nonlinear partial differential equations are solved by means of this method, a series of new exact solutions are obtained for the Fitzhugh-Nagumo equation. It can be accounted for that this method is one of the most effective approaches to seek the exact solutions of the nonlinear evolution equations, especially for nonintegrable models. The research paper based on this part-"New exact solutions to the Fitzhugh-Nagumo equation" has been accepted by "Applied Mathematics and Computation". We also obtain four groups of exact solutions for the Fisher equation by this method, and one of them is same to the "trial function solution" found by Liu, and we receive more results by this method at the same time.In chapterâ…¢, the three most commonly used methods, including classical similarity reduction, nonclassical similarity reduction and CK direct method are introduced. By applying these methods to a nonlinear dispersive-dissipative equation respectively, several different types of similarity reductions and a new similarity solution are given. It can be seen that these three methods are both similarities and characteristics.In chapterâ…£, the ARS method which is used to judge the Painleve property of the ODE and the WTC method which is used to judge the Painleve property of the PDE are specialized. Making the KdV equation as an example, we introduce how to find the Backlund transformation of a partial differential equation by using the Painleve analysis. Applying the Painleve PDE Test to a nonlinear dispersive-dissipative equation, we fred that the equation does not possess the Painleve property, and therefore it is not to be completely integrable.