Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model
Abstract
1. Introduction
- If Then, the model (1) is reduced to the weighted generalized Hattaf fractional model.
- If and Then, the model (1) is reduced to the Atangana–Baleanu fractional model.
- If Then, the model (1) is reduced to the weighted Atangana–Baleanu fractional model.
- If and Then, the model (1) is reduced to the Caputo–Fabrizio fractional model.
- Presenting a rigorous analysis of the existence and uniqueness of solutions for nonlinear hybrid fractional differential equations using a novel PFD within a p-Laplacian context which has not been extensively studied.
- Offering a generalized model that encompasses several existing formulations by varying a tuning power parameter.
- Demonstrating the Hyers–Ulam stability of the proposed model, indicating the robustness of the solutions under small perturbations.
- Providing numerical simulations for a range of cases and showing the application of the model to a real-world application through a complex disease transmission model.
- Ultimately, our findings provide an alternative framework for modeling complex systems with memory processes, creating opportunities for more sophisticated and accurate modeling tools and new avenues for research into the applications of fractional calculus.
2. Basic Concepts
- represents the PML function given by
- represents a normalization positive function obeying
- denotes the standard weighted Riemann–Liouville fractional integral of order δ given by
- (i)
- (ii)
Hypothesis
3. Qualitative Behavior of the Power Nonlocal Model (1) with p-Laplacian Operator
3.1. Equivalent Integral Equation
3.2. Notations
3.3. Lipschitz Properties of Operator
3.4. Compactness of Operator
3.5. Existence of Solution
3.6. Uniqueness of Solution
3.7. Symmetric Cases of System (1)
- Case 1: If Then, the model (1) is reduced to the following the weighted generalized Hattaf fractional model
- Case 2: If and Then, the model (1) is reduced to the following Atangana–Baleanu fractional model
- Case 3: If Then, the model (1) is reduced to the following weighted Atangana–Baleanu fractional model
- Case 4: If and Then, the model (1) is reduced to the following Caputo–Fabrizio fractional model
3.8. Hyers–Ulam Stability
3.9. UH Stability of Symmetric Cases
4. Numerical Scheme
5. Application of the Numerical Scheme to an HBV Model
6. Symmetric Cases
7. Discusion and Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Algolam, M.S.; Almalahi, M.A.; Suhail, M.; Muflh, B.; Aldwoah, K.; Hassan, M.; Islam, S. Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model. Fractal Fract. 2025, 9, 92. https://doi.org/10.3390/fractalfract9020092
Algolam MS, Almalahi MA, Suhail M, Muflh B, Aldwoah K, Hassan M, Islam S. Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model. Fractal and Fractional. 2025; 9(2):92. https://doi.org/10.3390/fractalfract9020092
Chicago/Turabian StyleAlgolam, Mohamed S., Mohammed A. Almalahi, Muntasir Suhail, Blgys Muflh, Khaled Aldwoah, Mohammed Hassan, and Saeed Islam. 2025. "Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model" Fractal and Fractional 9, no. 2: 92. https://doi.org/10.3390/fractalfract9020092
APA StyleAlgolam, M. S., Almalahi, M. A., Suhail, M., Muflh, B., Aldwoah, K., Hassan, M., & Islam, S. (2025). Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model. Fractal and Fractional, 9(2), 92. https://doi.org/10.3390/fractalfract9020092