The Atangana-Fabrizio Hybrid Fractional Operator: Formulation, Analysis, and Application

Abstract

In this paper, we introduce the Atangana-Fabrizio (AF) fractional operator as a novel hybrid operator designed to bridge the gap between power-law and exponential-decay memory kernels in complex system modeling. The AF operator, governed by a hybridization parameter λ, is formulated as a convex combination of the Caputo-Fabrizio (CF) and Atangana-Baleanu-Caputo (ABC) operators. We establish its fundamental mathematical properties, including the Laplace transform, linearity, and the conditions defining the associated fractional integral, while also verifying the operator's validity with several examples. Existence and uniqueness of solutions to nonlinear AF-type fractional systems are ensured through Banach and Schaefer fixed-point theorems, and both Ulam-Hyers (UH) and Generalized Ulam-Hyers (GUH) stability results are obtained. Two illustrative examples are provided to validate and clarify the theoretical analysis. Finally, we apply the AF operator to a fractional-order drug release model governed by a Michaelis-Menten-type equation and develop a generalized Adams-Bashforth Method (ABM). Simulations demonstrate the influence of the hybridization parameter λ on the dynamics.