The Modified Homotopy Perturbation Method And Its Application To Nonlinear Partial Differential Equations

In this paper, we introduce the basic idea of the homotopy perturbation method and its revision processes in detail, and summarize its application in nonlinear science systematically, particularly in the aspect of solving the nonlinear partial differential equations.This paper is organized as follows:chapter one summarizes some main methods of finding the exact solutions and approximate solutions of the nonlinear partial dif-ferential equations brought forward from home and abroad, presents the background and their application operating process of the homotopy perturbation method and the modified homotopy perturbation methods, and introduces the aim of research and the primary contents of this paper briefly.Chapter two applies the homotopy perturbation method to solve a system of variant RLW equations, and obtains some new approximate solutions. Then simu-lates their images and analyzes the absolute errors between the exact and approximate solutions obtained to determine the precision of the approximate solutionsChapter three Applies the homotopy perturbation method to obtain the approxi-mate solutions of the ZK-BBM equation, meanwhile simulates their images and ana-lyzes their absolute errors between the exact solutions and the approximate solutions obtained.Chapter four obtains some exact and approximate solutions of the generalized Zakharov equation by using the modified homotopy perturbation method, and sim-ulates their images and analyzes their absolute errors of the exact and approximate solutions obtained.Chapter five tries to modify the homotopy perturbation method and applies it to solve the coupled Sine-Gordon equations, obtains some approximate solutions, and indicates this modified method is suited to solving a class of equations by comparing the absolute errors to the solution obtained by the adomian decomposition method.Chapter six transforms inhomogeneous nonlinear partial differential equation into a homogeneous equation, and then gets exact solutions without "noise terms" by using the homotopy perturbation method. Chapter seven gets the general form solutions of k(m,n) equation by using pade approximation and Laplace transform to the truncated series solutions obtained by the homotopy perturbation method under the help of Maple.Finally, we make a summary of this paper and look ahead of research orientation in future.