Approximate Solutions of Fractional Differential Equations Using Optimal q-Homotopy Analysis Method: A Case Study of Abel Differential Equations

Abstract

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations.

Keywords

• Abel differential equation
• optimal q-homotopy analysis method
• Caputo fractional derivative
• fractional differential transform method