Piecewise implicit coupled system under ABC fractional differential equations with variable order

Saleh S. Redhwan; Maoan Han; Mohammed A. Almalahi; Maryam Ahmed Alyami; Mona Alsulami; Najla Alghamdi; Saleh S. Redhwan; Maoan Han; Mohammed A. Almalahi; Maryam Ahmed Alyami; Mona Alsulami; Najla Alghamdi

doi:10.3934/math.2024743

AIMS Mathematics

2024, Volume 9, Issue 6: 15303-15324. doi: 10.3934/math.2024743

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Research article

Piecewise implicit coupled system under ABC fractional differential equations with variable order

1.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, China
2.
Department of Mathematics, Hajjah University, Hajjah, Yemen
3.
Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah, Saudi Arabia

Received: 23 February 2024 Revised: 13 April 2024 Accepted: 16 April 2024 Published: 28 April 2024
MSC : 34A08, 34A12, 47H10, 97M70

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research.
- ABC-fractional differential equation,
- nonlocal conditions,
- fixed point theorem
Citation: Saleh S. Redhwan, Maoan Han, Mohammed A. Almalahi, Maryam Ahmed Alyami, Mona Alsulami, Najla Alghamdi. Piecewise implicit coupled system under ABC fractional differential equations with variable order[J]. AIMS Mathematics, 2024, 9(6): 15303-15324. doi: 10.3934/math.2024743

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Abstract

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research.

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