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Symmetry
Volume 18
Issue 5
10.3390/sym18050862
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Article

On Some Incommensurate Fractional-Order Reaction–Diffusion Systems: The Degn–Harrison and Its Stability

by
Omar Kahouli
1,*,
Amel Hioual
2,
Adel Ouannas
2,3,
Waleed Mohammed Abdelfattah
4,5,
Younès Bahou
6,
Ilyes Abidi
7,
Sameir Hamed
1,
Mohamed Chaabane
8 and
Sarra Elgharbi
9
1
Department of Electronics Engineering, Applied College, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia
2
Department of Mathematics and Computer Science, University of Oum El Bouaghi, Oum El Bouaghi 04000, Algeria
3
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
4
College of Engineering, University of Business and Technology, Jeddah 23435, Saudi Arabia
5
Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt
6
Computer Science Department, Applied College, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia
7
Department of Management Information Systems, Applied College, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia
8
Department of Electrical Engineering, National Engineering School of Sfax (ENIS), University of Sfax, Sfax 3038, Tunisia
9
Chemistry Department, College of Science, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2026, 18(5), 862; https://doi.org/10.3390/sym18050862 (registering DOI)
Submission received: 3 April 2026 / Revised: 14 May 2026 / Accepted: 15 May 2026 / Published: 19 May 2026
(This article belongs to the Special Issue Symmetry in Mathematical and Statistical Sciences: Theory, Methods, and Applications)
Versions Notes

Abstract

In this paper, we consider a reaction–diffusion system governed by incommensurate fractional time derivatives based on the Degn–Harrison model. Its formulation incorporates various memory effects on axial position through Caputo derivatives of variable orders, producing a more realistic modeling of the temporal dynamics. This paper starts with a study of the spatially homogeneous system and establishes conditions for local stability by using the Matignon criterion. The spectral decomposition method under Neumann boundary condition is then applied to study the complete reaction–diffusion system and describe diffusion-induced instabilities. Our results indicate that the noninteger fractional orders lead to significant changes in stability regions, as well as the initiation of pattern formation. Specifically, the orders of fractions induced as a control variable are regarded to be effective in controlling the stability of the system, thus they are global (or positive) control variables when their values achieved at some levels apply to the entire saturation, etc. Our numerical simulations are in excellent agreement with the theoretical predictions and show that memory asymmetry induces complex spatiotemporal dynamics not seen for classical integer-order systems.
Keywords: incommensurate fractional-order systems; Caputo operator; Degn–Harrison model; reaction–diffusion systems; local asymptotic stability; nonlinear biochemical dynamics; numerical simulations incommensurate fractional-order systems; Caputo operator; Degn–Harrison model; reaction–diffusion systems; local asymptotic stability; nonlinear biochemical dynamics; numerical simulations

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MDPI and ACS Style

Kahouli, O.; Hioual, A.; Ouannas, A.; Abdelfattah, W.M.; Bahou, Y.; Abidi, I.; Hamed, S.; Chaabane, M.; Elgharbi, S. On Some Incommensurate Fractional-Order Reaction–Diffusion Systems: The Degn–Harrison and Its Stability. Symmetry 2026, 18, 862. https://doi.org/10.3390/sym18050862

AMA Style

Kahouli O, Hioual A, Ouannas A, Abdelfattah WM, Bahou Y, Abidi I, Hamed S, Chaabane M, Elgharbi S. On Some Incommensurate Fractional-Order Reaction–Diffusion Systems: The Degn–Harrison and Its Stability. Symmetry. 2026; 18(5):862. https://doi.org/10.3390/sym18050862

Chicago/Turabian Style

Kahouli, Omar, Amel Hioual, Adel Ouannas, Waleed Mohammed Abdelfattah, Younès Bahou, Ilyes Abidi, Sameir Hamed, Mohamed Chaabane, and Sarra Elgharbi. 2026. "On Some Incommensurate Fractional-Order Reaction–Diffusion Systems: The Degn–Harrison and Its Stability" Symmetry 18, no. 5: 862. https://doi.org/10.3390/sym18050862

APA Style

Kahouli, O., Hioual, A., Ouannas, A., Abdelfattah, W. M., Bahou, Y., Abidi, I., Hamed, S., Chaabane, M., & Elgharbi, S. (2026). On Some Incommensurate Fractional-Order Reaction–Diffusion Systems: The Degn–Harrison and Its Stability. Symmetry, 18(5), 862. https://doi.org/10.3390/sym18050862

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