Theoretical analysis of time fractal fractional pantograph stochastic differential equations
The fractal-fractional derivative, a significant mathematical concept that merges fractal geometry with fractional calculus, has garnered increasing attention for modeling complex systems. To the best of our knowledge, no existing work has addressed the well-posedness, regularity, and averaging prin...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-09-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818125001858 |
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| author | Muhammad Imran Liaqat Muhammad Bilal Riaz Ali Akgül |
| author_facet | Muhammad Imran Liaqat Muhammad Bilal Riaz Ali Akgül |
| author_sort | Muhammad Imran Liaqat |
| collection | DOAJ |
| description | The fractal-fractional derivative, a significant mathematical concept that merges fractal geometry with fractional calculus, has garnered increasing attention for modeling complex systems. To the best of our knowledge, no existing work has addressed the well-posedness, regularity, and averaging principle for fractal-fractional pantograph stochastic differential equations (FFrPSDEs). In this study, we fill this gap by presenting results under the Atangana fractal-fractional derivative with the Riemann–Liouville (RL) definition and a power-law kernel. These equations capture essential features such as fractal behavior, memory effects, nonlocal dynamics, stochasticity, and time delays. We first establish the existence and uniqueness of solutions using a fixed-point approach. Next, we present results on continuous dependence and solution regularity. We also prove an averaging principle that simplifies the analysis of complex systems. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical findings. |
| format | Article |
| id | doaj-art-68ab71a7870d42c2b7f1abb783a4378a |
| institution | Kabale University |
| issn | 2666-8181 |
| language | English |
| publishDate | 2025-09-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-68ab71a7870d42c2b7f1abb783a4378a2025-08-20T03:41:26ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812025-09-011510125810.1016/j.padiff.2025.101258Theoretical analysis of time fractal fractional pantograph stochastic differential equationsMuhammad Imran Liaqat0Muhammad Bilal Riaz1Ali Akgül2Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan; Corresponding author.IT4Innovations, VSB-Technical University of Ostrava, Ostrava, Czech Republic; Applied Science Research Center, Applied Science Private University, Amman, JordanDepartment of Electronics and Communication Engineering, Saveetha School of Engineering, SIMATS, Chennai, Tamilnadu, India; Siirt University, Art and Science Faculty, Department of Mathematics, 56100, Siirt, Turkey; Department of Computer Engineering, Biruni University, 34010 Topkapı, Istanbul, Turkey; Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, Nicosia/Mersin 10, PC: 99138, TurkeyThe fractal-fractional derivative, a significant mathematical concept that merges fractal geometry with fractional calculus, has garnered increasing attention for modeling complex systems. To the best of our knowledge, no existing work has addressed the well-posedness, regularity, and averaging principle for fractal-fractional pantograph stochastic differential equations (FFrPSDEs). In this study, we fill this gap by presenting results under the Atangana fractal-fractional derivative with the Riemann–Liouville (RL) definition and a power-law kernel. These equations capture essential features such as fractal behavior, memory effects, nonlocal dynamics, stochasticity, and time delays. We first establish the existence and uniqueness of solutions using a fixed-point approach. Next, we present results on continuous dependence and solution regularity. We also prove an averaging principle that simplifies the analysis of complex systems. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical findings.http://www.sciencedirect.com/science/article/pii/S2666818125001858Averaging principleFixed point approachFractal-fractional derivativeWell-posedness |
| spellingShingle | Muhammad Imran Liaqat Muhammad Bilal Riaz Ali Akgül Theoretical analysis of time fractal fractional pantograph stochastic differential equations Partial Differential Equations in Applied Mathematics Averaging principle Fixed point approach Fractal-fractional derivative Well-posedness |
| title | Theoretical analysis of time fractal fractional pantograph stochastic differential equations |
| title_full | Theoretical analysis of time fractal fractional pantograph stochastic differential equations |
| title_fullStr | Theoretical analysis of time fractal fractional pantograph stochastic differential equations |
| title_full_unstemmed | Theoretical analysis of time fractal fractional pantograph stochastic differential equations |
| title_short | Theoretical analysis of time fractal fractional pantograph stochastic differential equations |
| title_sort | theoretical analysis of time fractal fractional pantograph stochastic differential equations |
| topic | Averaging principle Fixed point approach Fractal-fractional derivative Well-posedness |
| url | http://www.sciencedirect.com/science/article/pii/S2666818125001858 |
| work_keys_str_mv | AT muhammadimranliaqat theoreticalanalysisoftimefractalfractionalpantographstochasticdifferentialequations AT muhammadbilalriaz theoreticalanalysisoftimefractalfractionalpantographstochasticdifferentialequations AT aliakgul theoreticalanalysisoftimefractalfractionalpantographstochasticdifferentialequations |