Advanced Numerical and Semi-Analytical Methods for Differential Equations

12Homotopy Perturbation Method

12.1 Introduction

Homotopy perturbation method (HPM) is a semi‐analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. The HPM was proposed by J. He in 1999 [1]. This method was developed by making use of artificial parameters [2]. Interested readers may go through Refs. [3–6] for further details.

Almost all traditional perturbation methods are based on small parameter assumption. But, a majority of nonlinear problems have no small parameters at all and the determination of small parameters seems to be a special art requiring special techniques. These small parameters are so sensitive, such that a small change in small parameters will affect the results. An appropriate choice of small parameters leads to ideal results. However, an unsuitable choice of small parameters results in bad effects, sometimes seriously. Liu [2] proposed artificial parameter method and Liao [7,8] contributed homotopy analysis method to eliminate small parameter assumption. Further, He [ 1 ,9] developed two effective techniques viz. variational iteration method (VIM) and HPM in which no small parameter assumptions are required, where details of the VIM are given in Chapter 13.

12.2 Basic Idea of HPM

In this section we illustrate the basic idea of the HPM. For this we consider the following differential equation:

(12.1)

subject ...