Numerical Simulation of Delayed Memory Effects in Aging via Discrete Generalized ABC-Fractional Models

This study develops a novel discrete fractional aging model based on the Atangana–Baleanu–Caputo (ABC) derivative with a generalized Mittag–Leffler (GMLF) kernel. The proposed framework incorporates both memory-driven dynamics and delay-dependent degradation, enabling the simulation of realistic physiological or mechanical aging processes. We introduce a discrete ABC–GMLF operator and apply it to time-evolving systems subject to variable delays and nonlinear external stimuli. Numerical simulations are carried out using Gaussian quadrature and Simpson’s rule to approximate the generalized kernel, with special attention to convergence and stability. The impact of the fractional order α, the memory parameters (μ, ν, κ), and delay structure τ (n) is explored in detail. Through a series of synthetic and real-data experiments, we demonstrate how adjusting memory strength and delay mechanisms captures diverse aging responses, from smooth monotonic decay to oscillatory, nonlinear behaviors. The framework is further validated using parameter estimation via the L-BFGS-B algorithm and error analysis. This work lays the foundation for advanced modeling of aging processes with hereditary, adaptive, and feedback characteristics. A real-data validation based on WHO aging statistics is included, confirming the practical applicability of the proposed framework.