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

Junsheng Duan () and Lixia Jing ()
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Junsheng Duan: Shanghai Institute of Technology
Lixia Jing: Shanghai Institute of Technology

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 489-498

Abstract: 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.

Keywords: Fractional calculus; Fractional diffusion equation; Chebyshev polynomials; Mittag-Leffler function; Collocation method; 34A08; 26A33; 35R11; 41A50 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00495-y

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