THEORETICAL STUDY OF FRACTIONAL DIFFERENTIAL EQUATIONS UNDER UNCERTAINTY INVOLVING GENERAL MEMORY KERNELS

This study examines the theoretical and practical aspects of uncertain fractional-order differential equations, which model complex systems characterized by memory effects and human-based uncertainty. These equations provide a flexible and accurate representation of real-world phenomena, particularly when traditional probabilistic methods fall short. The analytical solution is derived using the Mittag-Leffler function within the framework of the Caputo–Katugampola derivative. Furthermore, the existence and uniqueness of the solution are established through a successive approximation approach, demonstrating sample continuity under Lipschitz and linear growth conditions. Finally, to illustrate the application, we consider a model of interest rates and provide two numerical examples to demonstrate the theoretical results on existence and uniqueness. The results are presented in a generalized form: when [Formula: see text], they reduce to the Caputo derivative, and as [Formula: see text], they correspond to the Caputo–Hadamard fractional derivative.