Computational Fractional Dynamical Systems

13Homotopy Analysis Method

13.1 Introduction

In Chapter 12, we have discussed the residual power series method (RPSM) based on the generalized Taylor’s series formula and the residual error function for solving linear and nonlinear ordinary/partial/fractional differential equations. In this chapter, we will address the homotopy analysis method (HAM). Various computationally efficient methods have been developed recently to solve fractional differential equations. However, neither perturbation nor non‐perturbation approaches can provide an easy way to conveniently modify and monitor the convergence region and rate of the series. Liao (1995, 2005b) suggested the homotopy analysis approach in 1992. Based on the homotopy of topology, the validity of the HAM is independent of whether there exist small parameters in the considered equation (Liao 2003, 2004, 2009) or not. Hence, HAM can overcome the previous restrictions and limitations of the perturbation techniques to allow us to examine highly nonlinear problems (Liao 2005a). This approach includes a certain auxiliary parameter ℏ ≠ 0 and an auxiliary linear operator L, which gives us an easy way to monitor and adjust the rate of convergence of the series solution (Jena et al. 2019; Srivastava et al. 2020). Several researchers have successfully applied HAM to solve various types of nonlinear physical problems arising in science and engineering (Zhang et al. 2011; Sakar and Erdogan 2013).

13.2 Theory of Homotopy Analysis Method ...