A computational method for fractional equations using Legendre polynomials and Mittag-Leffler kernels

In this paper, we present a novel numerical method for solving fractional-order differential equations and fractional integro-differential equations involving the Atangana-Baleanu-Caputo (ABC) fractional derivative with a generalized three-parameter Mittag-Leffler kernel. In contrast to previous approaches that approximate the solution itself, our method directly approximates the fractional derivative using shifted Legendre polynomials. This strategy results in a remarkably low residual error. The accuracy and reliability of the method are demonstrated through two examples: a fractional integro-differential equation modeling population dynamics in a closed system, and the fractional Blasius boundary value problem. The approximate solutions are illustrated in figures that show close align with the results reported in earlier studies. Additionally, we provide rigorous proof of convergence for the utilized method.