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A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

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Abstract

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Wenjie Liu

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Li-Lian Wang

  3. Mathematics and Statistics, Central South University, Changsha, 410083, China

    Shuhuang Xiang

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  1. Wenjie Liu
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  2. Li-Lian Wang
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Correspondence to Li-Lian Wang.

Additional information

W. Liu: The research of this author is partially supported by the China Postdoctoral Science Foundation Funded Project (No. 2017M620113), the National Natural Science Foundation of China (Nos. 11801120, 71773024 and 11771107), the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF.2019058) and the Natural Science Foundation of Heilongjiang Province of China (No. G2018006).

L. Wang: The research of this author is partially supported by Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059).

S. Xiang: This work of this author is partially supported by National Science Foundation of China (No. 11371376) and the Innovation-Driven Project and Mathematics.

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Liu, W., Wang, LL. & Xiang, S. A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations. Commun. Appl. Math. Comput. 1, 207–230 (2019). https://doi.org/10.1007/s42967-019-00012-1

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  • DOI: https://doi.org/10.1007/s42967-019-00012-1

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