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An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains

Author

Listed:
  • Laurence Finch

    (Program in Computational Analysis and Modeling, Louisiana Tech University, Ruston, LA 71272, USA)

  • Weizhong Dai

    (Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USA)

  • Aniruddha Bora

    (Division of Applied Mathematics, Brown University, Providence, RI 02906, USA)

Abstract

The simulation of wave propagation, such as soliton propagation, based on the Rosenau-KdV-RLW equation on unbounded domains requires a bounded computational domain. Therefore, a special boundary treatment, such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML), is necessary to minimize the reflections of outgoing waves at the boundary, preventing interference with the simulation’s accuracy. However, the presence of higher-order partial derivatives, such as u x x t and u x x x x t in the Rosenau-KdV-RLW equation, raises challenges in deriving accurate artificial boundary conditions. To address this issue, we propose an artificial neural network (ANN) method that enables soliton propagation through the computational domain without imposing artificial boundary conditions. This method randomly selects training points from the bounded computational space-time domain, and the loss function is designed based solely on the initial conditions and the Rosenau-KdV-RLW equation itself, without any boundary conditions. We analyze the convergence of the ANN solution theoretically. This new ANN method is tested in three examples. The results indicate that the present ANN method effectively simulates soliton propagation based on the Rosenau-KdV-RLW equation in unbounded domains or over extended periods.

Suggested Citation

  • Laurence Finch & Weizhong Dai & Aniruddha Bora, 2025. "An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains," Mathematics, MDPI, vol. 13(7), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1036-:d:1618248
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    References listed on IDEAS

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    1. Jinsong Hu & Yulan Wang, 2013. "A High-Accuracy Linear Conservative Difference Scheme for Rosenau-RLW Equation," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-8, November.
    2. Laila F. Seddek & Essam R. El-Zahar & Jae Dong Chung & Nehad Ali Shah, 2023. "A Novel Approach to Solving Fractional-Order Kolmogorov and Rosenau–Hyman Models through the q-Homotopy Analysis Transform Method," Mathematics, MDPI, vol. 11(6), pages 1-11, March.
    3. Jun Zhang & Zixin Liu & Fubiao Lin & Jianjun Jiao, 2019. "Asymptotic Analysis and Error Estimate for Rosenau-Burgers Equation," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-8, June.
    4. Wenchao Deng & Beibei Wu, 2022. "Numerical solution of Rosenau–KdV equation using Sinc collocation method," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 33(10), pages 1-17, October.
    5. He, Dongdong & Pan, Kejia, 2015. "A linearly implicit conservative difference scheme for the generalized Rosenau–Kawahara-RLW equation," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 323-336.
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