Criteria for Finite-Time Convergence in Discrete Variable-Order Fractional FitzHugh–Nagumo Reaction–Diffusion Systems

Abstract

This study addresses the problem of finite-time stability (FTS) for a discrete-time FitzHugh–Nagumo reaction–diffusion system (FHN–RDs) governed by a variable-order (VO) Caputo fractional difference operator. The discrete fractional formulation is obtained by combining a central difference approximation for the spatial derivative with a VO fractional operator for the temporal derivative. The analysis begins with proving the well-posedness of solutions for the proposed discrete model. The main contribution lies in establishing an FTS criterion. By employing a discrete fractional Gronwall-type inequality, we derive a sufficient stability condition expressed through the discrete Mittag–Leffler function (MLF). Finally, a numerical simulation is provided to illustrate the applicability of the theoretical findings, confirming that the system state remains bounded within a prescribed limit over a finite time horizon.