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: where , , , and are constants, and is the Rieman–Liouville fractional partial derivative of order . Two new implicit numerical methods with convergence order and for the fractional cable equation are proposed, respectively, where and 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.
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e-mail: f.liu@qut.edu.au
e-mail: q.yang@qut.edu.au
e-mail: i.turner@qut.edu.au
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January 2011
Research Papers
Two New Implicit Numerical Methods for the Fractional Cable Equation
Fawang Liu,
Fawang Liu
School of Mathematical Sciences,
e-mail: f.liu@qut.edu.au
Queensland University of Technology
, GPO Box 2434, Brisbane QLD 4001, Australia
Search for other works by this author on:
Qianqian Yang,
Qianqian Yang
School of Mathematical Sciences,
e-mail: q.yang@qut.edu.au
Queensland University of Technology
, GPO Box 2434, Brisbane QLD 4001, Australia
Search for other works by this author on:
Ian Turner
Ian Turner
School of Mathematical Sciences,
e-mail: i.turner@qut.edu.au
Queensland University of Technology
, GPO Box 2434, Brisbane QLD 4001, Australia
Search for other works by this author on:
Fawang Liu
School of Mathematical Sciences,
Queensland University of Technology
, GPO Box 2434, Brisbane QLD 4001, Australiae-mail: f.liu@qut.edu.au
Qianqian Yang
School of Mathematical Sciences,
Queensland University of Technology
, GPO Box 2434, Brisbane QLD 4001, Australiae-mail: q.yang@qut.edu.au
Ian Turner
School of Mathematical Sciences,
Queensland University of Technology
, GPO Box 2434, Brisbane QLD 4001, Australiae-mail: i.turner@qut.edu.au
J. Comput. Nonlinear Dynam. Jan 2011, 6(1): 011009 (7 pages)
Published Online: October 4, 2010
Article history
Received:
July 11, 2009
Revised:
November 11, 2009
Online:
October 4, 2010
Published:
October 4, 2010
Citation
Liu, F., Yang, Q., and Turner, I. (October 4, 2010). "Two New Implicit Numerical Methods for the Fractional Cable Equation." ASME. J. Comput. Nonlinear Dynam. January 2011; 6(1): 011009. https://doi.org/10.1115/1.4002269
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