Homotopy Perturbation Method For Nonlinear Differential Equations And Approximate Homotopy Direct Reduction Method

Nonlinear science has been extensively applied in the fields of physics, chemistry, engineering technology and economics. etc and trends towards flourishing with progress of the further research on the discipline, for which, it is crucial to investigate nonlinear differential equations that are usually used to dipict the nonlinear phenomenons and to pursue their solutions. Along with studies in depth, perturbed partial differential equations turns to be a main subject.The theory of perturbation, with using a small parameter, is capable to solve the physical systems which hardly can obtain exact solutions so as to be valued in physics and mathematics. etc. Nevertheless, it is still very difficult to solve the equations which involve the parameters not small enough or even have no perturbation. In order to settle the problem above-mentioned researchers have discovered that it was effective of conbining homotopy which derived from topology with traditional technique for solving equations. Thus, the two types of methods concerned with homotopy, namely, homotopy analysis method and homotopy perturbation method are presented. The approximal solution to the equation will be achieved with the traditional perturbation technique after building a model of homotopy for the given equation, introducing a homotopy parameter which could be considered as a small parameter.The methods of homotopy analysis and homotopy perturbation have great advantages in application for many scientific fields. Furthermore, one could derive more powerful reduction method when conbining them with additional methods. The chapter four of this thesis is to combine the homotopy analysis method with CK direct method to construct approximate homotopy direct reduction method. Relying on this method, we conclusively could obtain the approximate solution under building a homotopy model and taking the CK direct method and perturbation technique.This thesis will be divided into the chapters following:The first chapter is to introduce the background of the research on the theor of homotopy and the related knowledge.The second chapter is to investigate the damping KdV equation and the mKdV equation through the homotopy perturbed method, and to achieve the approximate solutions to the equations.The third part is to discuss the Nizhnik-Norikov-Vesselow equation using the extended homotopy perturbed method and to solve it reductionally.The forth part is to apply the approximate homotopy direct reduction method to the damping KdV equation, and to solve it in reduction.The last part is to summerize the whole thesis and propose several problems which need to be settled down.