Delay differential equations with fractional differential operators: Existence, uniqueness and applications to chaos

İrem Akbulut Arık; Seda İğret Araz; İrem Akbulut Arık; Seda İğret Araz

doi:10.3934/cam.2024008

Communications in Analysis and Mechanics

2024, Volume 16, Issue 1: 169-192. doi: 10.3934/cam.2024008

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Delay differential equations with fractional differential operators: Existence, uniqueness and applications to chaos

İrem Akbulut Arık ¹,
Seda İğret Araz ^{1,2
,
,}

1.
Faculty of Education, Department of Mathematics and Science Education, Siirt University, Turkey
2.
Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa

Received: 29 September 2023 Revised: 30 November 2023 Accepted: 22 January 2024 Published: 01 February 2024
34A12, 34C28, 34Kxx

In this study, we consider a chaotic model in which fractional differential operators and the delay term are added. Using the Carathéodory existence-uniqueness theorem for this chaotic model modified with the Caputo fractional derivative, we show that the solution of the associated system exists and is unique. We consider the chaotic model with a delay term with Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives and present a numerical algorithm for these models. We then present the numerical solution of chaotic models with delay terms by using piecewise differential operators, where fractional, classical and stochastic processes can be used. We present the numerical solution of chaotic models with delay terms, as modified by using piecewise differential operators. The graphical representations of these models are simulated for different values of the fractional order.
- Carathéodory existence-uniqueness theorem,
- chaotic models,
- delay term,
- piecewise derivative
Citation: İrem Akbulut Arık, Seda İğret Araz. Delay differential equations with fractional differential operators: Existence, uniqueness and applications to chaos[J]. Communications in Analysis and Mechanics, 2024, 16(1): 169-192. doi: 10.3934/cam.2024008

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Abstract

In this study, we consider a chaotic model in which fractional differential operators and the delay term are added. Using the Carathéodory existence-uniqueness theorem for this chaotic model modified with the Caputo fractional derivative, we show that the solution of the associated system exists and is unique. We consider the chaotic model with a delay term with Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives and present a numerical algorithm for these models. We then present the numerical solution of chaotic models with delay terms by using piecewise differential operators, where fractional, classical and stochastic processes can be used. We present the numerical solution of chaotic models with delay terms, as modified by using piecewise differential operators. The graphical representations of these models are simulated for different values of the fractional order.

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