The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains

Abstract

This study uses Laguerre-Hermite spectral Galerkin and spectral collocation methods for solving time-fractional sub-diffusion equations on unbounded domains. In the time domain, the generalized associated Laguerre functions of the first kind are employed as basis functions. In the Galerkin method, the fractional derivative of the basis functions can be obtained using the Laplace transform and its inverse. In the collocation method, the solution is expanded in terms of suitable global basis functions, and collocation conditions are imposed on the Gauss points. The corresponding errors are estimated in the time-space domain, and numerical examples verify the results and provide additional insight into the convergence behavior of the proposed method.

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Funding

This work is partially supported by the National Natural Science Foundation of China (Grant No. U1637208), National Key Research and Development Program of China (2017YFB1401801), Natural Scientific Foundation of Heilongjiang Province in China (A2016003), and Research Project of China Scholarship Council (No. 201706120212).

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Correspondence to Dazhi Zhang.

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Yu, H., Wu, B. & Zhang, D. The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains. Numer Algor 82, 1221–1250 (2019). https://doi.org/10.1007/s11075-018-00652-z

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Keywords

  • Spectral Galerkin method
  • Spectral collocation method
  • The generalized associated Laguerre functions
  • Hermite function
  • Time-fractional sub-diffusion equation
  • Unbounded domain