Numerical Algorithms

, Volume 72, Issue 2, pp 447–466 | Cite as

Galerkin finite element method and error analysis for the fractional cable equation

Original Paper

Abstract

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+hl+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ2+hl+1).

Keywords

Galerkin finite element approximation Fractional cable equation Semi-discrete finite difference approximation Full discretization Error analysis 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  3. 3.Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputationXiamen UniversityXiamenChina

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