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Crank-Nicolson-weighted-shifted-Grünwald-difference schemes for space Riesz variable-order fractional diffusion equations

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Abstract

In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for space Riesz variable-order fractional diffusion equations. Based on weighted-shifted-Grünwald-difference (WSGD) operators proposed in Lin and Liu (J. Comput. Appl. Math. 363, 77–91 (2020)) for Riemann-Liouville fractional derivatives, we derive WSGD operators for variable-order ones by using the relation between variable-order fractional derivative and (constant-order) fractional derivative. We then apply Crank-Nicolson-weighted-shifted-Grünwald-difference (CN-WSGD) schemes to the initial-boundary problem for space Riesz variable-order diffusion equations. Theoretical results on the stability and convergence of CN-WSGD schemes are presented and proved. Moreover, we derive a problem-based method to choose suitable CN-WSGD schemes, which leads to unconditioned stable linear systems with optimal upper bound for accuracy. Numerical results show that the proposed schemes are very efficient.

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Acknowledgments

We thank the referees for providing valuable comments and suggestions, which are very helpful for improving our paper.

Funding

This work is supported by National Natural Science Foundation of China (11771265) and the research grants MYRG2016-00077-FST, MYRG2019-00042-FST from University of Macau.

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, People’s Republic of China

    Fu-Rong Lin & Qiu-Ya Wang

  2. Department of Mathematics, University of Macau, Macao, China

    Xiao-Qing Jin

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  1. Fu-Rong Lin
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Correspondence to Fu-Rong Lin.

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Lin, FR., Wang, QY. & Jin, XQ. Crank-Nicolson-weighted-shifted-Grünwald-difference schemes for space Riesz variable-order fractional diffusion equations. Numer Algor 87, 601–631 (2021). https://doi.org/10.1007/s11075-020-00980-z

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