0
Research Papers

Two New Implicit Numerical Methods for the Fractional Cable Equation

[+] Author and Article Information
Fawang Liu1

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australiaf.liu@qut.edu.au

Qianqian Yang

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australiaq.yang@qut.edu.au

Ian Turner

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australiai.turner@qut.edu.au

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011009 (Oct 04, 2010) (7 pages) doi:10.1115/1.4002269 History: Received July 11, 2009; Revised November 11, 2009; Published October 04, 2010; Online October 04, 2010

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with fractional order temporal operators have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, we consider the following fractional cable equation involving two fractional temporal derivatives: u(x,t)/t=D0t1γ1(κ(2u(x,t)/x2))μ02Dt1γ2u(x,t)+f(x,t), where 0<γ1, γ2<1, κ>0, and μ02 are constants, and D0t1γu(x,t) is the Rieman–Liouville fractional partial derivative of order 1γ. Two new implicit numerical methods with convergence order O(τ+h2) and O(τ2+h2) for the fractional cable equation are proposed, respectively, where τ and h are the time and space step sizes. The stability and convergence of these methods are investigated using the energy method. Finally, numerical results are given to demonstrate the effectiveness of both implicit numerical methods. These techniques can also be applied to solve other types of anomalous subdiffusion problems.

FIGURES IN THIS ARTICLE
<>
Copyright © 2011 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Plot of u(x,t) versus x for γ1=γ2=0.5 (solid line) and γ1=γ2=1.0 (dotted line) at different times T=0.1,0.5

Grahic Jump Location
Figure 2

Plot of u(x,t) versus x for fixed γ1=0.1 and different γ2=0.2,0.4,0.6,0.8,1.0 at time T=0.1. γ2 increases in the direction of the arrow.

Grahic Jump Location
Figure 3

Plot of u(x,t) versus x for fixed γ2=0.8 and different γ1=0.2,0.4,0.6,0.8,1.0 at time T=0.1. γ1 increases in the direction of the arrow.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In