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Fast Numerical Contour Integral Method for Fractional Diffusion Equations

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Abstract

The numerical contour integral method with hyperbolic contour is exploited to solve space-fractional diffusion equations. By making use of the Toeplitz-like structure of spatial discretized matrices and the relevant properties, the regions that the spectra of resulting matrices lie in are derived. The resolvent norms of the resulting matrices are also shown to be bounded outside of the regions. Suitable parameters in the hyperbolic contour are selected based on these regions to solve the fractional diffusion equations. Numerical experiments are provided to demonstrate the efficiency of our contour integral methods.

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Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The first author was supported by the National Natural Science Foundation of China under Grant 11201192, the Natural Science Foundation of Jiangsu Province under Grant BK2012577, and the Natural Science Foundation for Colleges and Universities in Jiangsu Province under Grant 12KJB110004. The second author was supported by research grants 005/2012/A1 from FDCT of Macao and MYRG206(Y3-L4)-FST11-SHW from University of Macau.

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  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Hong-Kui Pang

  2. Department of Mathematics, University of Macau, Macao, People’s Republic of China

    Hai-Wei Sun

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  1. Hong-Kui Pang
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  2. Hai-Wei Sun
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Correspondence to Hai-Wei Sun.

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Pang, HK., Sun, HW. Fast Numerical Contour Integral Method for Fractional Diffusion Equations. J Sci Comput 66, 41–66 (2016). https://doi.org/10.1007/s10915-015-0012-9

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  • DOI: https://doi.org/10.1007/s10915-015-0012-9

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