Direct Algebraic Method And The Approximate Homotopy Direct Reduction Method Of Application

Recently, with the substantial study and application of linear system in the field of natural science, nonlinear system is widely developed and has become the research priority in many branches of natural science.As the study of nonlinear systems deepens, it is inevitably that people have to deal with a wide variety of nonlinear partial differential equations (NPDEs) that describe nonlinear phenomena. Correspondingly, seeking solutions to the NPDEs and the discussions on characteristics of these solutions have now become the essential subjects in Nonlinear System research.In the process of research, considering the effects of factors such as dissipa-tion, dispersion and outside drive, a plenty of questions come down to perturbed NPDEs, of which the reductions and approximate solutions required to be solved give rise to methods for study of perturbation. An approximate homotopy direct reduction method (AHDRM) has been proposed and applied to many perturbed nonlinear systems, which is a combination of homotopy concept, perturbation theory, CK direct reduction method and homotopy analysis method.With the aid of computer algebra system Maple, this paper is devoted to using the direct algebra method and AHDRM to solve some NPDEs and perturbed NPDEs. Our work is somewhere between the following aspects:Chapter one gives a brief review of the backgrounds of nonlinear systems; introduces several methods to solve perturbed nonlinear systems; and introduces the homotopy model and AHDRM.Chapter two shows the exact solutions for NPDEs.Chapter three reveals the reductions and infinite series solutions to the perturbed nonlinear systems. Finally, a summarization of this paper is presented to bring up some topics requiring further discussion.