Analyzing Coupled Delayed Fractional Systems: Theoretical Insights and Numerical Approaches
Abstract
:1. Introduction
- and are the unknown functions and .
- denotes the CapFD of order with .
- and are given continuous functions.
- , , , and are given continuous delay functions satisfying,and .
- Theoretical Analysis: We establish the existence, uniqueness, and Ulam–Hyers stability criteria for the coupled system of CapFDEs with delays.
- Numerical Validation: We implement the Adams–Bashforth–Moulton method to numerically approximate solutions, reinforcing our theoretical results.
- Application: We demonstrate the practical applicability of our findings through a variety of illustrative examples, applications, and numerical simulations.
2. Preliminaries
3. Main Results
- (H1)
- For all and , we have the following:
- and are continuous.
- and are bounded: There exist constants such that
- (H2)
- The delay functions satisfy the following:
- for all .
- for all .
- (H3)
- are given continuous initial functions.
- (H4)
- For all and , and satisfy Lipschitz conditions, with constants such that for all ,
- Let . We show that is relatively compact via the Arzelá–Ascoli theorem.
- For , is constant and hence equicontinuous. By Arzelá–Ascoli theorem, is compact.
4. Examples
- For ,
- For ,
- For ,
- For ,
- Lipschitz constants: .
- Stability constant K:
- Error bound for an -approximate solution with :
5. Applications
5.1. Control Systems with Communication Delays
- The time domain is discretized into N steps of size .
- Predictor step: An initial estimate of the solution is calculated using the Adams–Bashforth fractional formula:
- Corrector step: The estimate is refined using the Adams–Moulton fractional formula:
- Delay handling: Delayed terms (e.g., ) are handled by interpolating historical data at .
5.2. Predator–Prey Dynamics
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Arab, M.; Abdo, M.S.; Alghamdi, N.; Awadalla, M. Analyzing Coupled Delayed Fractional Systems: Theoretical Insights and Numerical Approaches. Mathematics 2025, 13, 1113. https://doi.org/10.3390/math13071113
Arab M, Abdo MS, Alghamdi N, Awadalla M. Analyzing Coupled Delayed Fractional Systems: Theoretical Insights and Numerical Approaches. Mathematics. 2025; 13(7):1113. https://doi.org/10.3390/math13071113
Chicago/Turabian StyleArab, Meraa, Mohammed S. Abdo, Najla Alghamdi, and Muath Awadalla. 2025. "Analyzing Coupled Delayed Fractional Systems: Theoretical Insights and Numerical Approaches" Mathematics 13, no. 7: 1113. https://doi.org/10.3390/math13071113
APA StyleArab, M., Abdo, M. S., Alghamdi, N., & Awadalla, M. (2025). Analyzing Coupled Delayed Fractional Systems: Theoretical Insights and Numerical Approaches. Mathematics, 13(7), 1113. https://doi.org/10.3390/math13071113