Finite-Time Stability Analysis of a Discrete-Time Generalized Reaction–Diffusion System

Othman Abdullah Almatroud and Adel Ouannas ()
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Othman Abdullah Almatroud: Department of Mathematics, College of Science, University of Hail, Hail 2240, Saudi Arabia
Adel Ouannas: Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04112, Algeria

Mathematics, 2024, vol. 12, issue 23, 1-17

Abstract: This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab’s LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems.

Keywords: generalized reaction–diffusion system; finite difference methods; discrete system; time delays; Lengyel–Epstein system; Matlab’s LMI toolbox; finite-time stability; Degn–Harrison system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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