Stability Analysis of Fractional-Order Nonlinear Alcohol Consumption Model and Numerical Simulation
Abstract
1. Introduction
- The alcohol addiction model is studied within a closed environment.
- Economic status and gender do not influence the risk of alcohol addiction.
- All participants interact with each other uniformly and to the same degree.
- The alcohol consumption of addicted individuals can influence those who are not yet addicted.
- Alcohol use plays a significant role in the spread of depression among addicted individuals, with personal differences either mitigating or intensifying this effect.
- Insufficient research has focused on the analysis of the Hyers–Ulam stability, which leads to a gap in the existing literature. Our primary objective is to examine the Hyers–Ulam stability of a fractional-order alcohol model.
- The methodology employs a fixed point approach to examine the existence and uniqueness solution, as well as investigate the Hyers–Ulam stability and highlight other interesting findings related to the stability of this system, which have not been previously explored in this context.
- In addition, this approach simplifies the calculation of approximate solutions for the given problem, enabling us to test and examine the theoretical results using real-world scenarios.
2. Model Description for the Alcohol Model
3. Preliminaries
4. Equilibrium Points of Alcohol Model
- Here, .
5. Existence and Uniqueness Results
- (A1) For the constants , such that
- (A2) For the constants £, with then,
- Step 1: In this step, we shall prove that the state variables can be displayed in terms of the continuity of the operator T as follows: Suppose that , for , .As a result, the continuity ∇, with the , consequently T is continuous.
- Step 2: In this step, we verify that this requirement for T is bounded. Assume that , the operator T, which satisfies the growth conditions as follows:We characterize to be a bounded subset of B and there exist we have . With the help of the growth condition, it follows that , in the same manner, such thatSo, we reach this goal as is bounded.
- Step 3: In addition, we prove that the continuity of T is continuous. For this purpose, let , such that , we have
- Step 4: Suppose that and , such that
6. Stability for Solution
- 1.
- .
- 2.
- = .
- 1.
- .
- 2.
- = .
7. Numerical Scheme
Results and Discussion
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Sivashankar, M.; Boulaaras, S.; Sabarinathan, S. Stability Analysis of Fractional-Order Nonlinear Alcohol Consumption Model and Numerical Simulation. Fractal Fract. 2025, 9, 61. https://doi.org/10.3390/fractalfract9020061
Sivashankar M, Boulaaras S, Sabarinathan S. Stability Analysis of Fractional-Order Nonlinear Alcohol Consumption Model and Numerical Simulation. Fractal and Fractional. 2025; 9(2):61. https://doi.org/10.3390/fractalfract9020061
Chicago/Turabian StyleSivashankar, Murugesan, Salah Boulaaras, and Sriramulu Sabarinathan. 2025. "Stability Analysis of Fractional-Order Nonlinear Alcohol Consumption Model and Numerical Simulation" Fractal and Fractional 9, no. 2: 61. https://doi.org/10.3390/fractalfract9020061
APA StyleSivashankar, M., Boulaaras, S., & Sabarinathan, S. (2025). Stability Analysis of Fractional-Order Nonlinear Alcohol Consumption Model and Numerical Simulation. Fractal and Fractional, 9(2), 61. https://doi.org/10.3390/fractalfract9020061