An efficient numerical technique for investigating the generalized Rosenau–KDV–RLW equation by using the Fourier spectral method

Shumoua F. Alrzqi; Fatimah A. Alrawajeh; Hany N. Hassan; Shumoua F. Alrzqi; Fatimah A. Alrawajeh; Hany N. Hassan

doi:10.3934/math.2024420

AIMS Mathematics

2024, Volume 9, Issue 4: 8661-8688. doi: 10.3934/math.2024420

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An efficient numerical technique for investigating the generalized Rosenau–KDV–RLW equation by using the Fourier spectral method

1.
Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia
2.
Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, Saudi Arabia

Received: 01 December 2023 Revised: 18 February 2024 Accepted: 23 February 2024 Published: 29 February 2024
MSC : 35C08, 65M06, 65T50

In this article, the generalized Rosenau-Korteweg-de Vries-regularized long wave (GR–KDV–RLW) equation was numerically studied by employing the Fourier spectral collection method to discretize the space variable, while the central finite difference method was utilized for the time dependency. The efficiency, accuracy, and simplicity of the employed methodology were tested by solving eight different cases involving four examples of the single solitary wave with different parameter values, interactions between two solitary waves, interactions between three solitary waves, and evolution of solitons with Gaussian and undular bore initial conditions. The error norms were evaluated for the motion of the single solitary wave. The conservation properties of the GR–KDV–RLW equation were studied by computing the momentum and energy. Additionally, the numerical results were compared with the previous studies in the literature.
- GR–KDV–RLW equation,
- Fourier spectral method,
- fast Fourier transform,
- finite difference,
- solitary waves
Citation: Shumoua F. Alrzqi, Fatimah A. Alrawajeh, Hany N. Hassan. An efficient numerical technique for investigating the generalized Rosenau–KDV–RLW equation by using the Fourier spectral method[J]. AIMS Mathematics, 2024, 9(4): 8661-8688. doi: 10.3934/math.2024420

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Abstract

In this article, the generalized Rosenau-Korteweg-de Vries-regularized long wave (GR–KDV–RLW) equation was numerically studied by employing the Fourier spectral collection method to discretize the space variable, while the central finite difference method was utilized for the time dependency. The efficiency, accuracy, and simplicity of the employed methodology were tested by solving eight different cases involving four examples of the single solitary wave with different parameter values, interactions between two solitary waves, interactions between three solitary waves, and evolution of solitons with Gaussian and undular bore initial conditions. The error norms were evaluated for the motion of the single solitary wave. The conservation properties of the GR–KDV–RLW equation were studied by computing the momentum and energy. Additionally, the numerical results were compared with the previous studies in the literature.

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