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Spectral element method with geometric mesh for two-sided fractional differential equations

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Abstract

Solutions of two-sided fractional differential equations (FDEs) usually exhibit singularities at the both endpoints, so it can not be well approximated by a usual polynomial based method. Furthermore, the singular behaviors are usually not known a priori, making it difficult to construct special spectral methods tailored for given singularities. We construct a spectral element approximation with geometric mesh, describe its efficient implementation, and derive corresponding error estimates. We also present ample numerical examples to validate our error analysis.

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Acknowledgments

Z.M. is supported in part by the MURI/ARO on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications” (W911NF-15-1-0562).

J.S is supported in part by AFOSR FA9550-16-1-0102 and NSF DMS-1620262, and by NSFC grants 11371298, 91630204 and 11421110001.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George St, Providence, RI, 02912, USA

    Zhiping Mao

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-1957, USA

    Jie Shen

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  1. Zhiping Mao
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  2. Jie Shen
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Correspondence to Jie Shen.

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Communicated by: Martin Stynes

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Mao, Z., Shen, J. Spectral element method with geometric mesh for two-sided fractional differential equations. Adv Comput Math 44, 745–771 (2018). https://doi.org/10.1007/s10444-017-9561-9

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  • DOI: https://doi.org/10.1007/s10444-017-9561-9

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