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

  • Wenjie Liu
  • Li-Lian WangEmail author
  • Shuhuang Xiang
Original Paper


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.


Fractional differential equations Generalised Birkhoff interpolation Nonstandard singular basis functions 

Mathematics Subject Classification

41A05 41A10 41A25 65M70 65N35 


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© Shanghai University 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  3. 3.Mathematics and StatisticsCentral South UniversityChangshaChina

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