On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations

Abstract

In this study, an optimal homotopy analysis approach will be described to deal with nonlinear fractional differential equations. The proposed approach presents a procedure to find an optimal auxiliary linear operator and the corresponding optimal initial approximation that will accelerate the convergence of series solutions for nonlinear differential equations with fractional derivatives. Then, a reliable modified version of the homotopy analysis method is presented to facilitate the calculations. Numerical comparison will be made to examine the computational efficiency and the pertinent features of the proposed algorithm. This algorithm is expected to be further employed to solve wide classes of nonlinear problems in fractional calculus.