Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative
This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated...
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MDPI AG
2025-06-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/7/411 |
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| author | N. S. Alharthi |
| author_facet | N. S. Alharthi |
| author_sort | N. S. Alharthi |
| collection | DOAJ |
| description | This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied. |
| format | Article |
| id | doaj-art-0968e8f8014f465ba3fedcb7d74c525a |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-0968e8f8014f465ba3fedcb7d74c525a2025-08-20T03:07:58ZengMDPI AGFractal and Fractional2504-31102025-06-019741110.3390/fractalfract9070411Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional DerivativeN. S. Alharthi0Department of Mathematics, Faculty of Sciences and Arts, King Abdulaziz University, Rabigh 21911, Saudi ArabiaThis work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied.https://www.mdpi.com/2504-3110/9/7/411Yang transformKdV–Burgers equationSawada–Kotera equationhomotopy perturbation methodAdomian decomposition methodCaputo operator |
| spellingShingle | N. S. Alharthi Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative Fractal and Fractional Yang transform KdV–Burgers equation Sawada–Kotera equation homotopy perturbation method Adomian decomposition method Caputo operator |
| title | Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative |
| title_full | Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative |
| title_fullStr | Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative |
| title_full_unstemmed | Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative |
| title_short | Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative |
| title_sort | dynamical and computational analysis of fractional korteweg de vries burgers and sawada kotera equations in terms of caputo fractional derivative |
| topic | Yang transform KdV–Burgers equation Sawada–Kotera equation homotopy perturbation method Adomian decomposition method Caputo operator |
| url | https://www.mdpi.com/2504-3110/9/7/411 |
| work_keys_str_mv | AT nsalharthi dynamicalandcomputationalanalysisoffractionalkortewegdevriesburgersandsawadakoteraequationsintermsofcaputofractionalderivative |