Coupled System of Variable-Order Fractional Differential Equations

This work explores the growing field of fractional calculus, with particular emphasis on the complexities and opportunities associated with variable-order derivatives. We critically assess existing definitions, identifying those that are consistent with the established principles of constant-order fractional calculus. Based on this analysis, we introduce new formulations derived from the Grünwald–Letnikov and Liouville approaches, together with a novel variable-order Mittag–Leffler function. The core of our study is devoted to investigating the existence and uniqueness of solutions for a coupled system of variable-order fractional differential equations subject to initial conditions. Using Schauder’s fixed-point theorem and the Banach contraction principle, we establish new results that contribute to strengthening the theoretical foundation of such dynamical systems.