The Mittag-Leffler–Caputo–Fabrizio Fractional Derivative and Its Numerical Approach

This study introduces a novel fractional-order derivative, termed the Mittag-Leffler–Caputo–Fabrizio (MLCF) fractional derivative, which is characterized by a singular kernel. Symmetry plays a key role in the structure and behavior of fractional operators, and our formulation reflects this by incorporating symmetric properties of the Mittag-Leffler function and its integral representation. To numerically approximate the MLCF derivative, we apply a two-point finite forward difference scheme to estimate the first-order derivative of the function $u\left(\lambda \right)$ within the integral component of the definition. This leads to the construction of a new numerical differentiation scheme. Our analysis demonstrates that the proposed approximation exhibits first-order convergence, with absolute errors decreasing as the time step size h diminishes. These errors are quantified by comparing our numerical results with exact analytical solutions, reinforcing the accuracy of the method.