Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives
This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Bale...
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MDPI AG
2025-02-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/2/104 |
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| author | F. Gassem Mohammed Almalahi Osman Osman Blgys Muflh Khaled Aldwoah Alwaleed Kamel Nidal Eljaneid |
| author_facet | F. Gassem Mohammed Almalahi Osman Osman Blgys Muflh Khaled Aldwoah Alwaleed Kamel Nidal Eljaneid |
| author_sort | F. Gassem |
| collection | DOAJ |
| description | This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. It uniquely features a tunable power parameter “<i>p</i>”, providing enhanced control over the representation of memory effects compared to traditional derivatives with fixed kernels. Utilizing the fixed-point theory, we rigorously establish the existence and uniqueness of solutions for these systems under appropriate conditions. Furthermore, we prove the Hyers–Ulam stability of the system, demonstrating its robustness against small perturbations. We complement this framework with a practical numerical scheme based on Lagrange interpolation polynomials, enabling efficient computation of solutions. Examples illustrating the model’s applicability, including symmetric cases, are supported by graphical representations to highlight the approach’s versatility. These findings address a significant gap in the literature and pave the way for further research in fractional calculus and its diverse applications. |
| format | Article |
| id | doaj-art-de2aa5ae62f142ddb6d59fb142ffa66c |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-de2aa5ae62f142ddb6d59fb142ffa66c2025-08-20T03:12:11ZengMDPI AGFractal and Fractional2504-31102025-02-019210410.3390/fractalfract9020104Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf DerivativesF. Gassem0Mohammed Almalahi1Osman Osman2Blgys Muflh3Khaled Aldwoah4Alwaleed Kamel5Nidal Eljaneid6Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi ArabiaDepartment of Artificial Intelligence, College of Computer and Information Technology, Al-Razi University, Sana’a 12544, YemenDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaThis paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. It uniquely features a tunable power parameter “<i>p</i>”, providing enhanced control over the representation of memory effects compared to traditional derivatives with fixed kernels. Utilizing the fixed-point theory, we rigorously establish the existence and uniqueness of solutions for these systems under appropriate conditions. Furthermore, we prove the Hyers–Ulam stability of the system, demonstrating its robustness against small perturbations. We complement this framework with a practical numerical scheme based on Lagrange interpolation polynomials, enabling efficient computation of solutions. Examples illustrating the model’s applicability, including symmetric cases, are supported by graphical representations to highlight the approach’s versatility. These findings address a significant gap in the literature and pave the way for further research in fractional calculus and its diverse applications.https://www.mdpi.com/2504-3110/9/2/104evolution systemspower non-local kernelsexistence of solutionsHyers–Ulam stability analysisfractional derivatives |
| spellingShingle | F. Gassem Mohammed Almalahi Osman Osman Blgys Muflh Khaled Aldwoah Alwaleed Kamel Nidal Eljaneid Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives Fractal and Fractional evolution systems power non-local kernels existence of solutions Hyers–Ulam stability analysis fractional derivatives |
| title | Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives |
| title_full | Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives |
| title_fullStr | Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives |
| title_full_unstemmed | Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives |
| title_short | Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives |
| title_sort | nonlinear fractional evolution control modeling via power non local kernels a generalization of caputo fabrizio atangana baleanu and hattaf derivatives |
| topic | evolution systems power non-local kernels existence of solutions Hyers–Ulam stability analysis fractional derivatives |
| url | https://www.mdpi.com/2504-3110/9/2/104 |
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