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The Galerkin finite element approximation of the fractional cable equation

Zhuang, P. , Liu, F., Anh, V., & Turner, I. (2012) The Galerkin finite element approximation of the fractional cable equation. In Cheng, Wen, Sun, HongGuang, & Baleanu, Dumitru (Eds.) The Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, Hohai University, Hohai University, Nanjing, pp. 1-8.

Abstract

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed.

The main contribution of this work is outlined as follow:

• A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the theoretical analysis.

Impact and interest:

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ID Code: 51450
Item Type: Conference Paper
Keywords: Fractional cable equation, finite difference, Galerkin finite element, stability, error estimation.
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000)
Australian and New Zealand Standard Research Classification > ENGINEERING (090000)
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright 2012 [please consult the author]
Deposited On: 06 Jul 2012 08:27
Last Modified: 13 Jun 2013 00:54

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