Initial value problem for fractional differential equations of variable order

Souad Guedim; Amar Benkerrouche; Mohammed Said Souid; Abdelkader Amara; Rashid Jan; Imtiaz Ahmad; Souad Guedim; Amar Benkerrouche; Mohammed Said Souid; Abdelkader Amara; Rashid Jan; Imtiaz Ahmad

doi:10.3934/mmc.2025026

Mathematical Modelling and Control

2025, Volume 5, Issue 4: 379-389. doi: 10.3934/mmc.2025026

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Research article

Initial value problem for fractional differential equations of variable order

1.
Laboratory of applied mathematics, Kasdi Merbah university, Ouargla, Algeria
2.
Faculty of exact science and computer science, University of Djelfa, P.O. Box 3117, Djelfa 17000, Algeria
3.
Department of Economic Sciences, University of Tiaret, Algeria
4.
Department of Mathematics, Kasdi Merbah university, Ouargla, Algeria
5.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
6.
Department of Mathematics, Khazar University, AZ1096, Baku Azerbaijan
7.
Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional (UNITEN), Kajang, Selangor 43000, Malaysia

Received: 17 May 2024 Revised: 17 September 2024 Accepted: 21 November 2024 Published: 19 November 2025

This study presented a novel approach to investigating the existence, uniqueness, and stability of solutions for an initial value problem involving fractional differential equations of variable order. In contrast to conventional methods in the literature, which often utilized generalized intervals and piecewise constant functions, we introduced a new fractional operator that is more appropriate for this problem. The existence and uniqueness of the solutions ware demonstrated through Leray-Schauder fixed point theorem and Banach's theorem, with an analysis of the uniform stability of the problem. The strength of our approach lies in its straightforwardness and reliance on fewer restrictive assumptions. The study concluded with an application that features a practical example, accompanied by visual illustrations.
- existence and uniqueness of solutions,
- initial value problem,
- fractional variable order,
- uniform stability,
- fixed point theorem
Citation: Souad Guedim, Amar Benkerrouche, Mohammed Said Souid, Abdelkader Amara, Rashid Jan, Imtiaz Ahmad. Initial value problem for fractional differential equations of variable order[J]. Mathematical Modelling and Control, 2025, 5(4): 379-389. doi: 10.3934/mmc.2025026

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Abstract

This study presented a novel approach to investigating the existence, uniqueness, and stability of solutions for an initial value problem involving fractional differential equations of variable order. In contrast to conventional methods in the literature, which often utilized generalized intervals and piecewise constant functions, we introduced a new fractional operator that is more appropriate for this problem. The existence and uniqueness of the solutions ware demonstrated through Leray-Schauder fixed point theorem and Banach's theorem, with an analysis of the uniform stability of the problem. The strength of our approach lies in its straightforwardness and reliance on fewer restrictive assumptions. The study concluded with an application that features a practical example, accompanied by visual illustrations.

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