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A fast algorithm for time-fractional diffusion equation with space-time-dependent variable order

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Abstract

We investigate a fast algorithm for the time-fractional diffusion equation with a space-time-dependent variable order, which models, e.g., the subdiffusion with varying memory effects. In addition to the traditional L1 discretization of the time-fractional derivative, we perform a further approximation for the L1 coefficients, analyze the structures of the resulting all-at-once system, and apply the divide and conquer method to obtain a fast numerical algorithm. Due to the spatial dependence of the variable order and the further approximation to the L1 coefficients, the temporal discretization coefficients are coupled with the inner product of the finite element method and lack the monotonicity, which are rarely encountered in previous works and thus motivate novel analysis methods and computational techniques. Compared with the standard time-stepping methods with L1 discretization, the proposed algorithm reduces the complexity of solving the all-at-once system from \(O(MN^2)\) to \(O(MN\ln ^3 N)\), where M stands for the spatial degree of freedom and N refers to the number of time steps. Numerical experiments are provided to substantiate the theoretical findings.

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Funding

This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 11971272 and 12001337, by the Natural Science Foundation of Shandong Province under Grant ZR2019BA026, and by the National Science Foundation under Grant No. DMS-2012291.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, China

    Jinhong Jia

  2. Department of Mathematics, University of South Carolina, Columbia, 29208, SC, USA

    Hong Wang

  3. School of Mathematics, Shandong University, Jinan, 250100, China

    Xiangcheng Zheng

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  1. Jinhong Jia
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  2. Hong Wang
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Xiangcheng Zheng and Jinhong Jia: methodology, writing—original draft. Hong Wang: conceptualization, review and editing.

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Correspondence to Xiangcheng Zheng.

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Jia, J., Wang, H. & Zheng, X. A fast algorithm for time-fractional diffusion equation with space-time-dependent variable order. Numer Algor 94, 1705–1730 (2023). https://doi.org/10.1007/s11075-023-01552-7

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