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Numerical methods for the forward and backward problems of a time-space fractional diffusion equation

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Abstract

In this paper, we consider the numerical methods for both the forward and backward problems of a time-space fractional diffusion equation. For the two-dimensional forward problem, we propose a finite difference method. The stability of the scheme and the corresponding Fast Preconditioned Conjugated Gradient algorithm are given. For the backward problem, since it is ill-posed, we use a quasi-boundary-value method to deal with it. Based on the Fourier transform, we obtain two kinds of order optimal convergence rates by using an a-priori and an a-posteriori regularization parameter choice rules. Numerical examples for both forward and backward problems show that the proposed numerical methods work well.

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Acknowledgements

The authors would like to offer their cordial thanks to the editorial board and the reviewers of this paper for their valuable comments and suggestions, without these suggestions there would be no present form of this paper.

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  1. School of Mathematics and Statistics, Xidian University, Xi’an, 713200, People’s Republic of China

    Xiaoli Feng, Xiaoyu Yuan & Meixia Zhao

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Zhi Qian

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  1. Xiaoli Feng
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The work is supported by the Natural Science Basic Research Program of Shaanxi (Program Nos. 2024JC-YBMS-034, 2024JC-YBQN-0050, 2023-JC-YB-054) and the National Natural Science Foundation of China (Nos. 61877046, 12071214).

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Feng, X., Yuan, X., Zhao, M. et al. Numerical methods for the forward and backward problems of a time-space fractional diffusion equation. Calcolo 61, 16 (2024). https://doi.org/10.1007/s10092-024-00567-3

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