Well-posedness and numerical approximation for nonlinear hybrid fractional differential equations with Caputo derivative

Abstract

This paper considers a class of nonlinear hybrid fractional differential equations with the Caputo derivative. The hybrid feature means that the fractional derivative is taken for the corrected state z(t)-f(t,z(t)), which can model memory effects together with a feedback-type term. We first convert the initial value problem into an equivalent Volterra integral equation. Using this form, we define an operator on C([0,T],R) and prove existence and uniqueness of solutions by the Banach fixed point theorem under simple Lipschitz conditions. We also establish well-posedness by showing continuous dependence on the initial data and deriving an explicit a priori bound. Finally, we develop a three-step Adams--Bashforth product-integration scheme for numerical approximation and provide numerical experiments to support the theoretical results.