Criteria of existence and stability of an n-coupled system of generalized Sturm-Liouville equations with a modified ABC fractional derivative and an application to the SEIR influenza epidemic model

Elkhateeb S. Aly; Mohammed A. Almalahi; Khaled A. Aldwoah; Kamal Shah; Elkhateeb S. Aly; Mohammed A. Almalahi; Khaled A. Aldwoah; Kamal Shah

doi:10.3934/math.2024691

AIMS Mathematics

2024, Volume 9, Issue 6: 14228-14252. doi: 10.3934/math.2024691

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Criteria of existence and stability of an n-coupled system of generalized Sturm-Liouville equations with a modified ABC fractional derivative and an application to the SEIR influenza epidemic model

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Kingdom of Saudi Arabia
2.
Nanotechnology Research Unit, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Kingdom of Saudi Arabia
3.
Department of Mathematics, Hajjah University, Hajjah, Yemen
4.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Al Madinah 42351, Saudi Arabia
5.
Department of Mathematics, University of Malakand, Chakdara Dir (Lower), 18000 Khyber Pakhtunkhwa, Pakistan
6.
Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon

Received: 29 February 2024 Revised: 09 April 2024 Accepted: 11 April 2024 Published: 18 April 2024
MSC : 34A08, 49J15, 65P99

The primary objective of this study was to explore the behavior of an n-coupled system of generalized Sturm-Liouville (GSL) and Langevin equations under a modified ABC fractional derivative. We aimed to analyze the dynamics of the system and gain insights into how this operator influences the conditions for the existence and uniqueness of solutions. We established the existence and uniqueness of solutions by employing the Banach contraction principle and Leray-Schauder's alternative fixed-point theorem. We also investigated the Hyers-Ulam stability of the system. This analysis allows us to understand the stability properties of the solutions and evaluate their sensitivity to perturbations. Furthermore, we employed Lagrange's interpolation polynomials to produce a numerical scheme for the influenza epidemic model. By combining theoretical analysis, mathematical principles, and numerical simulations, this study contributes to enriching our understanding of the behavior of the system and offers insights into its dynamics and practical applications in epidemiology.
- MABC fractional model,
- generalized Sturm-Liouville equation,
- stability analysis,
- Lagrange polynomials,
- graphical representation
Citation: Elkhateeb S. Aly, Mohammed A. Almalahi, Khaled A. Aldwoah, Kamal Shah. Criteria of existence and stability of an n-coupled system of generalized Sturm-Liouville equations with a modified ABC fractional derivative and an application to the SEIR influenza epidemic model[J]. AIMS Mathematics, 2024, 9(6): 14228-14252. doi: 10.3934/math.2024691

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Abstract

The primary objective of this study was to explore the behavior of an n-coupled system of generalized Sturm-Liouville (GSL) and Langevin equations under a modified ABC fractional derivative. We aimed to analyze the dynamics of the system and gain insights into how this operator influences the conditions for the existence and uniqueness of solutions. We established the existence and uniqueness of solutions by employing the Banach contraction principle and Leray-Schauder's alternative fixed-point theorem. We also investigated the Hyers-Ulam stability of the system. This analysis allows us to understand the stability properties of the solutions and evaluate their sensitivity to perturbations. Furthermore, we employed Lagrange's interpolation polynomials to produce a numerical scheme for the influenza epidemic model. By combining theoretical analysis, mathematical principles, and numerical simulations, this study contributes to enriching our understanding of the behavior of the system and offers insights into its dynamics and practical applications in epidemiology.

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