Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations
This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohan...
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| Format: | Article |
| Language: | English |
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2025-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/2/193 |
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| author | Khudhayr A. Rashedi Musawa Yahya Almusawa Hassan Almusawa Tariq S. Alshammari Adel Almarashi |
| author_facet | Khudhayr A. Rashedi Musawa Yahya Almusawa Hassan Almusawa Tariq S. Alshammari Adel Almarashi |
| author_sort | Khudhayr A. Rashedi |
| collection | DOAJ |
| description | This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohand variational iteration method (MVIM). The fractional derivatives in the equations are expressed using the Caputo operator, which provides a flexible framework for analyzing the dynamics of fractional systems. By leveraging these methods, we derive diverse types of solutions, including hyperbolic, trigonometric, and rational forms, illustrating the effectiveness of the techniques in addressing complex fractional models. Numerical simulations and graphical representations are provided to validate the accuracy and applicability of derived solutions. Special attention is given to the influence of the fractional parameter on behavior of the solution behavior, highlighting its role in controlling diffusion and wave propagation. The findings underscore the potential of q-HMTM and MVIM as robust tools for solving non-linear fractional differential equations. They offer insights into their practical implications in fluid dynamics, wave mechanics, and other applications governed by fractional-order models. |
| format | Article |
| id | doaj-art-849c4ae38e114ac692f9eb83c8cb419d |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-849c4ae38e114ac692f9eb83c8cb419d2025-01-24T13:39:42ZengMDPI AGMathematics2227-73902025-01-0113219310.3390/math13020193Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman EquationsKhudhayr A. Rashedi0Musawa Yahya Almusawa1Hassan Almusawa2Tariq S. Alshammari3Adel Almarashi4Deparment of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi ArabiaDeparment of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi ArabiaThis study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohand variational iteration method (MVIM). The fractional derivatives in the equations are expressed using the Caputo operator, which provides a flexible framework for analyzing the dynamics of fractional systems. By leveraging these methods, we derive diverse types of solutions, including hyperbolic, trigonometric, and rational forms, illustrating the effectiveness of the techniques in addressing complex fractional models. Numerical simulations and graphical representations are provided to validate the accuracy and applicability of derived solutions. Special attention is given to the influence of the fractional parameter on behavior of the solution behavior, highlighting its role in controlling diffusion and wave propagation. The findings underscore the potential of q-HMTM and MVIM as robust tools for solving non-linear fractional differential equations. They offer insights into their practical implications in fluid dynamics, wave mechanics, and other applications governed by fractional-order models.https://www.mdpi.com/2227-7390/13/2/193Sawada–Kotera equationRosenau–Hyman equationq-homotopy Mohand transform method (q-HMTM)Mohand variational iteration method (MVIM)fractional-order differential equationCaputo operator |
| spellingShingle | Khudhayr A. Rashedi Musawa Yahya Almusawa Hassan Almusawa Tariq S. Alshammari Adel Almarashi Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations Mathematics Sawada–Kotera equation Rosenau–Hyman equation q-homotopy Mohand transform method (q-HMTM) Mohand variational iteration method (MVIM) fractional-order differential equation Caputo operator |
| title | Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations |
| title_full | Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations |
| title_fullStr | Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations |
| title_full_unstemmed | Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations |
| title_short | Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations |
| title_sort | fractional dynamics applications of the caputo operator in solving the sawada kotera and rosenau hyman equations |
| topic | Sawada–Kotera equation Rosenau–Hyman equation q-homotopy Mohand transform method (q-HMTM) Mohand variational iteration method (MVIM) fractional-order differential equation Caputo operator |
| url | https://www.mdpi.com/2227-7390/13/2/193 |
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