Computational Fractional Dynamical Systems

5Homotopy Perturbation Method

5.1 Introduction

In this chapter, we will discuss about homotopy perturbation method (HPM), which is again a semi‐analytical approach for solving linear and nonlinear ordinary/partial/fractional differential equations. HPM was first proposed by He (1999a). This approach has been established using artificial parameters (Liu 1997). Interested readers may visit references (He 2003, 2004) for more information. Almost all conventional perturbation methods are based on the assumption of small parameters. However, most nonlinear problems have no small parameters, and the determination of small parameters needs a unique art requiring special techniques. These small parameters are so sensitive that a slight change may influence the final result. The right choice of small parameters yields optimal performance. However, an inappropriate choice of small parameters leads to poor, even significant effects. Liu (1997) proposed the artificial parameter method and Liao (1995, 1997) contributed to the homotopy analysis method to eradicate the presumption of small parameters. He (1999a, 1999b) also established a technique called the variational iteration method (VIM), in which no small parameter assumptions are made, and is discussed in Chapter 9.

In the subsequent sections, firstly, the theories behind the method with respect to fractional order are addressed. Then the systematic step‐by‐step procedure of the technique along with two problems are introduced.