Analytical study and numerical simulation of fractional differential equations within a unified operator framework

Abstract A general framework for Caputo-type fractional derivative operators with nonlocal and nonsingular kernels, together with their extensions that include integrable singular kernels and the associated fractional integral operators, was introduced. This generalized category includes many well-known operators that have been extensively utilized to model real-world problems in fractional calculus. This paper focuses on highlighting the advantages of the modified extensions that overcome the limitations of nonsingular kernel fractional derivatives. We investigate the existence and uniqueness of solutions to IVPs governed by fractional differential equations involving the generalized class of fractional derivatives and their extensions. One of the primary objectives of this work is to develop a numerical approach for simulating IVPs involving these operators. Using the proposed approach, L 1 -solutions are obtained for IVPs with extended fractional derivatives. To illustrate the applicability of the method, the dynamics of a fractional variant of the logistic model are analyzed as a case study.