Extension And Implementation Of The Homotopy Analysis Method

With the development of science and technology, more and more problems are de-scribed by nonlinear differential equations. People want to make precise quantitative analysis of these equations. In the last decades, the development of high performance computers and the achievement of symbolic computation system have pushed the study on analytic approximate solutions of differential equations. Quite a few good algorithms have proposed, such as the perturbation method, the Adomian decomposition method and the homotopy analysis method.The homotopy analysis method is a powerful algorithm for constructing analytic so-lutions of linear or nonlinear differential systems, which unifies the previous unperturbed methods, such as the artificial small parameter method, theδexpansion method and the Adomian decomposition method. Moreover, the homotopy analysis method itself pro-vides a tool to control and adjust the convergence rate and region of the solution series. This method has solved many strongly nonlinear problems.This article concentrates on the homotopy analysis method, and creates a condition to construct Rm, which gives some flexibility for the structuring of Rm by the homotopy analysis method. Besides, making use of Rach's study on the Adomian polynomials, we apply its structuring method to the homotopy analysis method, and provide several new Rm algorithms. We prove that the new formulae can get more expansions than the original method. Experiments also show that in some cases, these formulae can obtain more satisfactory solution.The second part of this article is to implement the homotopy analysis method. We develop a software package in computer algebraic system Maple, which involved the new defined Rm algorithms. The package can solve ordinary differential equations, partial differential equations and coupled systems, as well some special equations with undefined parameters and fractional derivative. The package used Block-based design ideas, which allows users to update the functions and expand its application.