Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation

Jinhong Jia; Xiangcheng Zheng; Hong Wang

doi:10.1515/fca-2021-0062

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Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation

Jinhong Jia , Xiangcheng Zheng und Hong Wang

Veröffentlicht/Copyright: 28. Oktober 2021

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Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 24 Heft 5

Abstract

We prove the wellposedness of a distributed-order space-fractional diffusion equation with variably distribution and its support, which could adequately model the challenging phenomena such as the anomalous diffusion in multiscale heterogeneous porous media, and smoothing properties of its solutions. We develop and analyze a collocation scheme for the proposed model based on the proved smoothing properties of the solutions. Furthermore, we approximately expand the stiffness matrix by a sum of Toeplitz matrices multiplied by diagonal matrices, which can be employed to develop the fast solver for the approximated system. We prove that it suffices to apply O(log N) terms of expansion to retain the accuracy of the numerical discretization of degree N, which reduces the storage of the stiffness matrix from O(N²) to O(N log N), and the computational cost of matrix-vector multiplication from O(N²) to O(N log² N). Numerical results are presented to verify the effectiveness and the efficiency of the fast method.

MSC 2010: 65F05; 65M70; 65R20; 26A33

Key Words and Phrases: distributed-order space-fractional diffusion equation; variably distribution; collocation method; fast method; Toeplitz matrix

Acknowledgements

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. This work was partially funded by the National Natural Science Foundation of China under Grants 11971272 and 12001337, by the ARO MURI Grant W911NF-15-1-0562, by the National Science Foundation under Grant DMS-2012291, by the China Postdoctoral Science Foundation 2021TQ0017, by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) YJ20210019, and by the Natural Science Foundation of Shandong Province under Grant ZR2019BA026.

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Received: 2020-12-20

Revised: 2021-08-29

Published Online: 2021-10-28

Published in Print: 2021-10-26

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https://doi.org/10.1515/fca-2021-0062

Schlagwörter für diesen Artikel

distributed-order space-fractional diffusion equation; variably distribution; collocation method; fast method; Toeplitz matrix