A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation

Muath Awadalla; Abdul Hamid Ganie; Dowlath Fathima; Adnan Khan; Jihan Alahmadi; Muath Awadalla; Abdul Hamid Ganie; Dowlath Fathima; Adnan Khan; Jihan Alahmadi

doi:10.3934/math.2024516

AIMS Mathematics

2024, Volume 9, Issue 5: 10561-10579. doi: 10.3934/math.2024516

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A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia; Email: mawadalla@kfu.edu.sa
2.
Basic Science Department, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia; Email: a.ganie@seu.edu.sa, d.fathima@seu.edu.sa
3.
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan; Email: adnanmummand@gmail.com
4.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam BinAbdulaziz University, Al-Kharj 11942, Saudi Arabia; Email: j.alahmadi@psau.edu.sa

Received: 26 January 2024 Revised: 08 March 2024 Accepted: 12 March 2024 Published: 18 March 2024
MSC : 26A33, 34A34, 35A22

The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $ \varsigma = 1 $, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.
- Caputo-Fabrizio operator,
- Yang transform,
- homotopy perturbation transform method
Citation: Muath Awadalla, Abdul Hamid Ganie, Dowlath Fathima, Adnan Khan, Jihan Alahmadi. A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation[J]. AIMS Mathematics, 2024, 9(5): 10561-10579. doi: 10.3934/math.2024516

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Abstract

The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $ \varsigma = 1 $, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.

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