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The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions

Author

Listed:
  • Ji Lin

    (College of Mechanics and Materials, Hohai University, Nanjing 210098, China
    These authors contributed equally to this work.)

  • Sergiy Reutskiy

    (A. Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine, 2/10 Pozharsky Street, 61046 Kharkiv, Ukraine
    These authors contributed equally to this work.)

  • Yuhui Zhang

    (College of Mechanics and Materials, Hohai University, Nanjing 210098, China
    These authors contributed equally to this work.)

  • Yu Sun

    (Nanjing Hydraulic Research Institute, Nanjing 210029, China
    These authors contributed equally to this work.)

  • Jun Lu

    (Nanjing Hydraulic Research Institute, Nanjing 210029, China
    These authors contributed equally to this work.)

Abstract

This article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second step is the meshless approximation where the Müntz polynomials are used to form the approximate solution and the unknown parameters are obtained by imposing the approximation for the governing equations. We illustrate first the detailed derivation of the analytic approximation and then the numerical implementation of the solution procedure. Several numerical examples are provided to verify the accuracy, efficiency, and adaptability to problems with general boundary conditions. The numerical results are compared with exact solutions and numerical methods reported in the literature, showing that the algorithm has great potential for multi-dimensional multi-term time-fractional equations with various boundary conditions.

Suggested Citation

  • Ji Lin & Sergiy Reutskiy & Yuhui Zhang & Yu Sun & Jun Lu, 2023. "The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions," Mathematics, MDPI, vol. 11(4), pages 1-26, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:929-:d:1065947
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    References listed on IDEAS

    as
    1. Želi, Velibor & Zorica, Dušan, 2018. "Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 2316-2335.
    2. Fardi, M. & Zaky, M.A. & Hendy, A.S., 2023. "Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 614-635.
    3. Fardi, Mojtaba & Khan, Yasir, 2021. "A novel finite difference-spectral method for fractal mobile/immobiletransport model based on Caputo–Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    4. Kamran Kamran & Zahir Shah & Poom Kumam & Nasser Aedh Alreshidi, 2020. "A Meshless Method Based on the Laplace Transform for the 2D Multi-Term Time Fractional Partial Integro-Differential Equation," Mathematics, MDPI, vol. 8(11), pages 1-14, November.
    5. Zhang, Hui & Jiang, Xiaoyun & Yang, Xiu, 2018. "A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 302-318.
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