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The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

Author

Listed:
  • Junsheng Duan

    (Shanghai Institute of Technology)

  • Lixia Jing

    (Shanghai Institute of Technology)

Abstract

In this paper, we consider the solution of the initial and boundary value problem for the time-space fractional diffusion equation in the sense of Caputo based on the Chebyshev collocation method. Firstly, the problem is converted to an initial value problem for a fractional integral-differential equation which absorbs the boundary conditions. Then the shifted Chebyshev polynomials and collocation method for the space variable are used. The coefficient functions of the Chebyshev expansion are solved through the Picard iterative process and the matrix Mittag-Leffler functions for the time variable. We also present a numerical method to cope with the improper convolution integral on the time variable. Finally, a numerical example is verified via the proposed method. The results demonstrate the effectiveness and great potential of the Chebyshev polynomials and the matrix Mittag-Leffler functions for the solution of the fractional differential equation.

Suggested Citation

  • Junsheng Duan & Lixia Jing, 2025. "The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method," Indian Journal of Pure and Applied Mathematics, Springer, vol. 56(2), pages 489-498, June.
    • Handle: RePEc:spr:indpam:v:56:y:2025:i:2:d:10.1007_s13226-023-00495-y
    DOI: 10.1007/s13226-023-00495-y
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    References listed on IDEAS

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    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
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