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Mathematics of Computation

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Finite difference/spectral approximations for the fractional cable equation
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by Yumin Lin, Xianjuan Li and Chuanju Xu PDF
Math. Comp. 80 (2011), 1369-1396 Request permission

Abstract:

The Cable equation has been one of the most fundamental equations for modeling neuronal dynamics. In this paper, we consider the numerical solution of the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is threefold: 1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order $O(\triangle t^{2-\max \{\alpha ,\beta \}}+ \triangle t^{-1}N^{-m})$, where $\triangle t,N$ and $m$ are respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. $\alpha$ and $\beta$ are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims.
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Additional Information
  • Yumin Lin
  • Affiliation: School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
  • Xianjuan Li
  • Affiliation: School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
  • Chuanju Xu
  • Affiliation: School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
  • Email: cjxu@xmu.edu.cn
  • Received by editor(s): February 7, 2009
  • Received by editor(s) in revised form: April 11, 2010
  • Published electronically: December 2, 2010
  • Additional Notes: The research of the first author was partially supported by Fujian NSF under Grant S0750017.
    The research of the third author was partially supported by National NSF of China under Grant 10531080, the Excellent Young Teachers Program by the Ministry of Education of China, and 973 High Performance Scientific Computation Research Program 2005CB321703
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1369-1396
  • MSC (2010): Primary 65M12, 65M06, 65M70, 35S10
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02438-X
  • MathSciNet review: 2785462