The Homotopy Analysis Method for Solving Differential Equations with Generalized Caputo-Type Fractional Derivatives

Abstract

This study expands and modifies the homotopy analysis method to handle differential equations with generalized Caputo-type fractional derivatives. Analytical approximate solutions for such models were successfully provided using the proposed modification. The determination of the valid region of convergence for the proposed method, with respect to the auxiliary control parameter, was discussed when using fractional operators. Then, mainly, the accuracy and effectiveness of the proposed method was verified through illustrative examples and comparisons with the predictor corrector method and RK4 method. Finally, it is expected that the studied generalized operators and the suggested method can be widely applied in the field of fractional calculus.