Work

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

Year: 2025

Type: article

Abstract: 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.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract9110710

Authors: Ali Turab, Josué-Antonio Nescolarde-Selva, Wajahat Ali, Andrés Montoyo, Jun-Jiat Tiang

Institutions: University of Alicante, Northwestern Polytechnical University, University of Ostrava, Multimedia University

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258