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A fast method for variable-order space-fractional diffusion equations

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Abstract

We develop a fast divide-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical scheme does not have a Toeplitz structure. In this paper, we derive a fast approximation of the coefficient matrix by the means of a finite sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires \(O(N\log ^{2} N)\) memory and \(O(N\log ^{3} N)\) computational complexity with N being the numbers of unknowns. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

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Acknowledgments

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Funding

This work was funded by the OSD/ARO MURI Grant W911NF-15-1-0562, by the National Science Foundation under grant DMS-1620194, by the National Natural Science Foundation of China (No. 11971482), by the Natural Science Foundation of Shandong Province (No. ZR2017MA006, No. ZR2019BA026), and by the China Scholarship Council (File No. 2018063-20326).

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Authors and Affiliations

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

    Jinhong Jia

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

    Xiangcheng Zheng & Hong Wang

  3. College of Science, China University of Petroleum, Qingdao, 266580, China

    Hongfei Fu

  4. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Pingfei Dai

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  1. Jinhong Jia
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  2. Xiangcheng Zheng
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  3. Hongfei Fu
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  4. Pingfei Dai
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  5. Hong Wang
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Correspondence to Hong Wang.

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Jia, J., Zheng, X., Fu, H. et al. A fast method for variable-order space-fractional diffusion equations. Numer Algor 85, 1519–1540 (2020). https://doi.org/10.1007/s11075-020-00875-z

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