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Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation

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Abstract

In this article, some second-order time discrete schemes covering parameter 𝜃 combined with Galerkin finite element (FE) method are proposed and analyzed for looking for the numerical solution of nonlinear cable equation with time fractional derivative. At time tk−𝜃, some second-order 𝜃 schemes combined with weighted and shifted Grünwald difference (WSGD) approximation of fractional derivative are considered to approximate the time direction, and the Galerkin FE method is used to discretize the space direction. The stability of second-order 𝜃 schemes is derived and the second-order time convergence rate in L2-norm is proved. Finally, some numerical calculations are implemented to indicate the feasibility and effectiveness for our schemes.

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Acknowledgments

The authors thank the reviewers and editor very much for their insightful comments for improving our work.

Funding

This work is supported by the National Natural Science Fund (11661058,11761053,11772046), Australian Research Council (ARC) via the Discovery Project (DP180103858) and Natural Science Fund of Inner Mongolia Autonomous Region (2016MS0102, 2017MS0107).

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Yang Liu, Yanwei Du, Hong Li & Yajun Wang

  2. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD, 4001, Australia

    Fawang Liu

  3. School of Mathematics, Jilin University, Changchun, 130012, China

    Yanwei Du

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  1. Yang Liu
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  2. Yanwei Du
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  3. Hong Li
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  4. Fawang Liu
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Correspondence to Fawang Liu.

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Liu, Y., Du, Y., Li, H. et al. Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation. Numer Algor 80, 533–555 (2019). https://doi.org/10.1007/s11075-018-0496-0

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  • DOI: https://doi.org/10.1007/s11075-018-0496-0

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