Computational and Parameter-Sensitivity Analysis of Dual-Order Memory-Driven Fractional Differential Equations with an Application to Animal Learning

Fractional differential equations are used to model complex systems where present dynamics depend on past states. In this work, we study a linear fractional model with two Caputo orders that captures long-term memory together with short-term adaptation. The existence and uniqueness of solutions are established using Banach and Krasnoselskii’s fixed-point theorems. A parameter study isolates the roles of the fractional orders and coefficients, yielding an explicit stability region in the (α,β)–plane via computable contraction bounds. For computation, we implement the Adams–Bashforth–Moulton (ABM) and fractional linear multistep (FLM) methods, comparing accuracy and convergence. As an application, we model animal learning in which proficiency evolves under memory effects and pulsed stimuli. The results quantify the impact of feedback timing on trajectories within the admissible region, thereby illustrating the suitability of dual-order fractional models for memory-driven behavior.