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A fast time stepping Legendre spectral method for solving fractional Cable equation with smooth and non-smooth solutions

Author

Listed:
  • Xu, Yibin
  • Liu, Yanqin
  • Yin, Xiuling
  • Feng, Libo
  • Wang, Zihua
  • Li, Qiuping

Abstract

To improve the calculation accuracy and efficiency, in this article, we develop a fast time stepping Legendre spectral method for solving fractional Cable equation, where in temporal direction the time stepping method is utilized and the spatial variable is discretized by Legendre spectral method. The time stepping method is used to approximate fractional order derivative, and its convergence accuracy in time is O(τ2). The fast algorithm is applied to the time stepping method and it can reduce the computational complexity from O(M2) to O(MlogM), where M denotes the number of time stepping. For non-smooth solutions, we deal with the initial singularity by adding correction terms. We also analyze the numerical stability and convergence in detail. Numerical experiments confirm our theoretical analysis and efficiency of the fast algorithm.

Suggested Citation

  • Xu, Yibin & Liu, Yanqin & Yin, Xiuling & Feng, Libo & Wang, Zihua & Li, Qiuping, 2023. "A fast time stepping Legendre spectral method for solving fractional Cable equation with smooth and non-smooth solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 154-170.
    • Handle: RePEc:eee:matcom:v:211:y:2023:i:c:p:154-170
    DOI: 10.1016/j.matcom.2023.04.009
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    References listed on IDEAS

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    1. Liu, Zeting & Lü, Shujuan & Liu, Fawang, 2018. "Fully discrete spectral methods for solving time fractional nonlinear Sine–Gordon equation with smooth and non-smooth solutions," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 213-224.
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