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Volume 7
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Article

Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model

by
Tareq Hamadneh
1,
Amel Hioual
2,*,
Omar Alsayyed
3,
Yazan Alaya AL-Khassawneh
4,
Abdallah Al-Husban
5 and
Adel Ouannas
6
1
Department of Mathematics, Faculty of Science, Al Zaytoonah University of Jordan, Amman 11733, Jordan
2
Laboratory of Dynamical Systems and Control, University of Oum EL-Bouaghi, Oum El Bouaghi 04000, Algeria
3
Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan
4
Data Science and Artificial Intelligence Department, Zarqa University, Zarqa 13110, Jordan
5
Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan
6
Department of Mathematics and Computer Science, University of Oum EL-Bouaghi, Oum El Bouaghi 04000, Algeria
*
Author to whom correspondence should be addressed.
Fractal Fract. 2023, 7(8), 587; https://doi.org/10.3390/fractalfract7080587
Submission received: 29 May 2023 / Revised: 2 July 2023 / Accepted: 24 July 2023 / Published: 29 July 2023
(This article belongs to the Special Issue Stability Analysis for Fractional-Order Equations)
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Abstract

In the last few years, reaction–diffusion models associated with discrete fractional calculus have risen in prominence in scientific fields, not just due to the requirement for numerical simulation but also due to the described biological phenomena. This work investigates a discrete equivalent of the fractional reaction–diffusion glycolysis model. The discrete fractional calculus tool is introduced to the discrete modeling of diffusion problems in the Caputo-like delta sense, and a fractional discretization diffusion model is described. The local stability of the equilibrium points in the proposed discrete system is examined. We additionally investigate the global stability of the equilibrium point by developing a Lyapunov function. Furthermore, this study indicates that the L1 finite difference scheme and the second-order central difference scheme can successfully preserve the characteristics of the associated continuous system. Finally, an equivalent summation representing the model’s numerical formula is shown. The diffusion concentration is further investigated for different fractional orders, and examples with simulations are presented to corroborate the theoretical findings.
Keywords: fractional discrete-time reaction–diffusion systems; L1 finite difference scheme; local stability; Lyapunov function; global stability fractional discrete-time reaction–diffusion systems; L1 finite difference scheme; local stability; Lyapunov function; global stability

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MDPI and ACS Style

Hamadneh, T.; Hioual, A.; Alsayyed, O.; AL-Khassawneh, Y.A.; Al-Husban, A.; Ouannas, A. Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model. Fractal Fract. 2023, 7, 587. https://doi.org/10.3390/fractalfract7080587

AMA Style

Hamadneh T, Hioual A, Alsayyed O, AL-Khassawneh YA, Al-Husban A, Ouannas A. Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model. Fractal and Fractional. 2023; 7(8):587. https://doi.org/10.3390/fractalfract7080587

Chicago/Turabian Style

Hamadneh, Tareq, Amel Hioual, Omar Alsayyed, Yazan Alaya AL-Khassawneh, Abdallah Al-Husban, and Adel Ouannas. 2023. "Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model" Fractal and Fractional 7, no. 8: 587. https://doi.org/10.3390/fractalfract7080587

APA Style

Hamadneh, T., Hioual, A., Alsayyed, O., AL-Khassawneh, Y. A., Al-Husban, A., & Ouannas, A. (2023). Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model. Fractal and Fractional, 7(8), 587. https://doi.org/10.3390/fractalfract7080587

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