scholarly journals An Explicit Numerical Method for the Fractional Cable Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
J. Quintana-Murillo ◽  
S. B. Yuste
Keyword(s):  
Stability Analysis ◽  
Numerical Method ◽  
Stability And Convergence ◽  
Cable Equation ◽  
Von Neumann ◽  
Fractional Equations ◽  
Forward Difference ◽  
Difference Formula ◽  
Fractional Cable Equation ◽  
Explicit Numerical Method

An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.

Stability and Convergence of Two New Implicit Numerical Methods for the Fractional Cable Equation

2009 ◽  
Author(s):  
F. Liu ◽  
Q. Yang ◽  
I. Turner
Keyword(s):  
Numerical Methods ◽  
Partial Derivative ◽  
Stability And Convergence ◽  
Cable Equation ◽  
Neuronal Dynamics ◽  
Neuronal Dendrites ◽  
Space Step ◽  
The Stability ◽  
Fundamental Equations ◽  
Fractional Cable Equation

The cable equation is one 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=0Dt1−γ1κ∂2u(x,t)∂x2−μ02Dt1−γ2u(x,t)+f(x,t), where 0 < γ1,γ2 < 1, κ > 0, and μ02 are constants, and 0Dt1−γu(x,t) is the Rieman-Liouville fractional partial derivative of order 1 − γ. 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.

Two New Implicit Numerical Methods for the Fractional Cable Equation

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Fawang Liu ◽  
Qianqian Yang ◽  
Ian Turner
Keyword(s):  
Numerical Methods ◽  
Partial Derivative ◽  
Stability And Convergence ◽  
Cable Equation ◽  
Neuronal Dynamics ◽  
Neuronal Dendrites ◽  
Space Step ◽  
The Stability ◽  
Fundamental Equations ◽  
Fractional Cable Equation

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=D0t1−γ1(κ(∂2u(x,t)/∂x2))−μ02Dt1−γ2u(x,t)+f(x,t), where 0<γ1, γ2<1, κ>0, and μ02 are constants, and D0t1−γu(x,t) is the Rieman–Liouville fractional partial derivative of order 1−γ. 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.

scholarly journals A new difference scheme for fractional cable equation and stability analysis

2015 ◽  
Vol 4 (1) ◽  
pp. 52 ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Difference Scheme ◽  
Caputo Fractional Derivative ◽  
Spectral Stability ◽  
Cable Equation ◽  
Unconditionally Stable ◽  
Fractional Cable Equation

<p>We consider the fractional cable equation. For solution of fractional Cable equation involving Caputo fractional derivative, a new difference scheme is constructed based on Crank Nicholson difference scheme. We prove that the proposed method is unconditionally stable by using spectral stability technique.</p>

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

2021 ◽  
Author(s):  
Avinash K. Mittal ◽  
Lokendra K. Balyan ◽  
Manoj K. Panda ◽  
Parnika Shrivastava ◽  
Harvindra Singh
Keyword(s):  
Cable Equation ◽  
Two Dimensional ◽  
Fractional Cable Equation
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