An Efficient Numerical Approach for Solving the Mittag-Leffler Fractional Differential Equations

In this paper, we introduce a fractional integral operator related to the recently proposed Mittag-Leffler-Caputo-Fabrizio (MLCF) fractional derivative, which has a non-singular Mittag-Leffler kernel. We provide sufficient conditions for the existence of unique solutions for a certain class of nonlinear fractional differential equations by fixed point methods. Our analysis yields an explicit inequality involving fractional orders, the Lipschitz constant, and the finite time, thus guaranteeing the existence of a unique solution. Furthermore, we introduce a novel numerical scheme, the Euler MLCF method, for approximating the solutions to these equations. We prove that this scheme is convergent with first-order accuracy. The new scheme is rigorously validated through numerous numerical examples. The results, in terms of absolute error and empirical convergence order, are consistent with the theoretical prediction. This paper presents theoretical and practical advancements in solving fractional differential equations using the MLCF operator.