The Comhined Research Of The Reproducing Kernel Method And Two Other Kinds Of Numerical Methods

Nowadays, the study of nonlinear mathematical physics problems has been the hot topics in the related research fields. However, it is hard for us to obtain the exact solutions to these questions. Therefore, the key point of this study is to solve and obtain the approximate solutions of them. To achieve these goals, the paper introduces three kinds of reliable and applicable numerical calculation methods, which can be used to solve the approximate solutions of nonlinear mathematical physics problems. These newly born numerical calculation methods are reproducing kernel method, Adomian decomposition methods, as well as the method of homotopy perturbation method.Based on the three numerical calculation methods mentioned above, the paper tentatively puts forward three kinds of numerical algorithms used to solve nonlinear mathematical physics problems. They are the improved Adomian decomposition method, the RKM-ADM decomposition methods and the RKM-HPM decomposition method. At the same time, the paper focuses on introducing the theories of the combined method. And we find that the new method can not only expands the scope of solution, but also has a relatively smaller amount of calculation and higher accuracy than the traditional methods.This thesis is arranged as follows:The first chapter is an introduction, which contains three sections. In this part, three kinds of numerical algorithms are introduced, and they are the reproducing kernel method, the Adomian decomposition methods as well as the homotopy perturbation method. Meanwhile, we briefly introduce the birth, the development and the present studied situation of the three kinds effective numerical algorithms.Chapter two is a brief introduction of these three kinds of numerical algorithms. First, we introduce the basic ideas, principles of Adomian decomposition methods, and the realization of its algorithm. Meanwhile, we put forward an improved Adomian decomposition method. Then, we discuss and analyze the basic theories of reproducing kernel method method. In the end of this part, the paper expounds the ideas of the theory and the algorithm of homotopy perturbation method.Chapter three introduces two kinds of combined methods, which are obtained from the combination of reproducing kernel method and two other kinds of methods mentioned above. The solving process of these two combined methods has been presented in details, and the methods are RKM-ADM decomposition method and the RKM-HPM decomposition method.Chapter four is the summary and outlook of this study. In this part, the author first makes a brief summary of the main work and research results of this study. Then, puts forward some suggestions and makes some prospects for the future studies.