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Finite Element Methods Based on Two Families of Second-Order Numerical Formulas for the Fractional Cable Model with Smooth Solutions

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Abstract

We apply two families of novel fractional \(\theta \)-methods, the FBT-\(\theta \) and FBN-\(\theta \) methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional \(\theta \)-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with \(O(\tau ^2+h^{r+1})\) for smooth solutions, where \(\tau \) is the time mesh size and h is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time.

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Acknowledgements

The authors are grateful to Professor Buyang Li, two anonymous referees and editors for their valuable suggestions which improve the presentation of this work. The work of the second author was supported in part by the NSFC grant 11661058. The work of the third author was supported in part by the NSFC grant 11761053, the NSF of Inner Mongolia 2017MS0107, and the program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region NJYT-17-A07. The work of the fourth author was supported in part by grants NSFC 11871092 and U1930402.

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Authors and Affiliations

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

    Baoli Yin, Yang Liu & Hong Li

  2. Beijing Computational Science Research Center, 100193, Beijing, China

    Zhimin Zhang

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

    Zhimin Zhang

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  1. Baoli Yin
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  2. Yang Liu
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Correspondence to Yang Liu.

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Yin, B., Liu, Y., Li, H. et al. Finite Element Methods Based on Two Families of Second-Order Numerical Formulas for the Fractional Cable Model with Smooth Solutions. J Sci Comput 84, 2 (2020). https://doi.org/10.1007/s10915-020-01258-1

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