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Spectral Methods for Substantial Fractional Differential Equations

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Abstract

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.

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Acknowledgements

We would like to thank Professor Martin Stynes for his valuable comments when preparing the manuscript.

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Authors and Affiliations

  1. School of Mathematical Science and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Fujian, 361005, China

    Can Huang

  2. Beijing Computational Science Research Center, Beijing, 100094, China

    Zhimin Zhang

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

  4. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Qingshuo Song

Authors
  1. Can Huang
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  2. Zhimin Zhang
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  3. Qingshuo Song
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Corresponding author

Correspondence to Can Huang.

Additional information

The research of Can Huang, is supported by National Natural Science Foundation of China under grant 11401500, 91630204 and the Fundamental Research Funds for the Central Universities under grant 20720150007.

The research of Zhimin Zhang, was supported in part by the National Natural Science Foundation of China under grants 11471031, 91430216, and the US National Science Foundation through grant DMS-1419040.

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Huang, C., Zhang, Z. & Song, Q. Spectral Methods for Substantial Fractional Differential Equations. J Sci Comput 74, 1554–1574 (2018). https://doi.org/10.1007/s10915-017-0506-8

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  • DOI: https://doi.org/10.1007/s10915-017-0506-8

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