Abstract
In this paper, a fast and memory-saving numerical scheme is presented for solving hidden-memory variable-order time-fractional diffusion equations based on the L1 method. Due to the nonlocality of fractional operators, the L1 method leads to a high computational complexity. To reduce the storage and computational cost, a modified exponential-sum-approximation method is utilized to approximate the convolution kernel involved in the fractional derivative. Additionally, one of the challenges faced during theoretical analysis is the loss of monotonicity of the temporal discretization coefficients caused by the hidden-memory variable order. A pioneering decomposition technique has been adopted to address this. The scheme has been theoretically proven to be convergent, and its effectiveness and accuracy have been confirmed through numerical examples.
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The data of codes involved in this paper are available from the corresponding author on reasonable request.
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Acknowledgements
This work is supported in part by research grants of the Science and Technology Development Fund, Macau SAR (file no. 0122/2020/A3), and University of Macau (file nos. MYRG-GRG2023-00085-FST-UMDF, MYRG-GRG2023-00181-FST-UMDF, MYRG2020-00208-FST and MYRG2022-00262-FST).
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Communicated by Roberto Garrappa.
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Sun, LY., Lei, SL. & Sun, HW. Efficient finite difference scheme for a hidden-memory variable-order time-fractional diffusion equation. Comp. Appl. Math. 42, 362 (2023). https://doi.org/10.1007/s40314-023-02504-6
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DOI: https://doi.org/10.1007/s40314-023-02504-6
Keywords
- Time-fractional equation
- Hidden-memory
- Variable-order
- Fast finite difference method
- Convergence analysis