Analytical and Numerical Analysis of a Memory-Dependent Fractional Model for Behavioral Learning Dynamics

Fractional differential equations offer a natural framework for describing systems in which present states are influenced by the past. This work presents a nonlinear Caputo-type fractional differential equation (FDE) with a nonlocal initial condition and attempts to describe a model of memory-dependent behavioral adaptation. The proposed framework uses a fractional-order derivative η∈(0,1) to discuss the long-term memory effects. The existence and uniqueness of solutions are demonstrated by Banach’s and Krasnoselskii’s fixed-point theorems. Stability is analyzed through Ulam–Hyers and Ulam–Hyers–Rassias benchmarks, supported by sensitivity results on the kernel structure and fractional order. The model is further employed for behavioral despair and learned helplessness, capturing the role of delayed stimulus feedback in shaping cognitive adaptation. Numerical simulations based on the convolution-based fractional linear multistep (FVI–CQ) and Adams–Bashforth–Moulton (ABM) schemes confirm convergence and accuracy. The proposed setup provides a compact computational and mathematical paradigm for analyzing systems characterized by nonlocal feedback and persistent memory.