Numerical solutions of fractional parabolic equations with generalized Mittag–Leffler kernels

Abstract

In this article, we investigate the generalized fractional operator Caputo type (ABC) with kernels of Mittag–Lefller in three parameters Eα,µγ(λ,t) and its fractional integrals with arbitrary order for solving the time fractional parabolic nonlinear equation. The generalized definition generates infinitely many problems for a fixed fractional derivative α. We utilize this operator with homotopy analysis method for constructing the new scheme for generating successive approximations. This procedure is used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the ℏ-curves as well as by calculating the residual error and the results were accurate. Based on the experiment, we verify the existence of the solution for the new parameters. Depending on these results, this treatment can be used to find approximate solutions to many fractional differential equations.