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The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

Author

Listed:
  • Peiyao Wang

    (School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China)

  • Shangwen Peng

    (School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China)

  • Yihao Cao

    (School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China)

  • Rongpei Zhang

    (School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China)

Abstract

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Suggested Citation

  • Peiyao Wang & Shangwen Peng & Yihao Cao & Rongpei Zhang, 2024. "The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform," Mathematics, MDPI, vol. 12(7), pages 1-14, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1110-:d:1371517
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    References listed on IDEAS

    as
    1. Chen, Zhong & Gou, QianQian, 2019. "Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 465-478.
    2. Nur Hasan Mahmud Shahen & Foyjonnesa & Md Habibul Bashar & Tasnim Tahseen & Sakhawat Hossain, 2021. "Solitary and Rogue Wave Solutions to the Conformable Time Fractional Modified Kawahara Equation in Mathematical Physics," Advances in Mathematical Physics, Hindawi, vol. 2021, pages 1-9, July.
    3. Liaqat, Muhammad Imran & Akgül, Ali, 2022. "A novel approach for solving linear and nonlinear time-fractional Schrödinger equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
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