Generalized Deformable Fractional Derivative via the Mittag-Leffler Function: Fundamental Properties, Integral Representation, and Applications

In this paper, we propose a new generalized form of the deformable fractional derivative constructed via the Mittag–Leffler function. The proposed operator extends both classical and existing fractional differentiation frameworks by introducing a flexible deformation parameter associated with the Mittag–Leffler kernel. Its main analytical properties are investigated, including linearity, product and chain rules, a generalized mean value theorem, and a Taylor-type expansion. A compatible fractional integral operator is also established, ensuring a coherent and unified structure within generalized fractional calculus. To assess the effectiveness of the proposed formulation, a numerical application was performed, which confirmed the theoretical results and demonstrated the consistency of the new deriva- tive with the classical case while capturing fractional dynamics. These findings highlight the potential of the proposed framework for modeling memory-dependent and complex dynamical systems.