Computational Fractional Dynamical Systems

15q‐Homotopy Analysis Method

15.1 Introduction

Chapter 13 has already discussed the homotopy analysis method (HAM), which is based on the coupling of the standard perturbation approach and homotopy in topology. This approach uses the auxiliary parameter ℏ, which gives us some versatility and great freedom to monitor and adjust the convergence region, including the convergence rate of the series solution. Subsequently, El‐Tawil and Huseen in 2012 (El‐Tawil and Huseen 2012) introduced an improvement to the HAM, which is called the q‐HAM. The q‐HAM is later used by Iyiola et al. (2013) to obtain a solution for the time‐fractional foams drainage equation. Iyiola (2013) constructed the numerical solution of the fifth‐order time‐fractional Ito and Sawada–Kotera equations using q‐HAM. The convergence of q‐HAM was considered in El‐Tawil and Huseen (2013). This method includes two auxiliary parameters n and ℏ which helps us to adjust and control the convergence of the solution. It may be noted that the standard HAM is obtained by replacing n = 1 in the q‐HAM. It is already mentioned that q‐HAM is an improved HAM scheme and does not require discretization, perturbation, or linearization. The introduction of the additional parameter n in the q‐HAM provides greater flexibility than the HAM in adjusting and controlling the convergence region and the convergence rate of the series solution. Several authors have recently used q‐HAM to solve (integer and non‐integer) linear and nonlinear differential ...