scholarly journals Numerical simulation of fractional Cable equation of spiny neuronal dendrites

2014 ◽  
Vol 5 (2) ◽  
pp. 253-259 ◽  
Author(s):  
N.H. Sweilam ◽  
M.M. Khader ◽  
M. Adel
Keyword(s):  
Numerical Simulation ◽  
Cable Equation ◽  
Neuronal Dendrites ◽  
Fractional Cable Equation

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.

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

scholarly journals Analytical Solution of Generalized Space-Time Fractional Cable Equation

Mathematics ◽  
2015 ◽  
Vol 3 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Ram Saxena ◽  
Zivorad Tomovski ◽  
Trifce Sandev
Keyword(s):  
Analytical Solution ◽  
Space Time ◽  
Cable Equation ◽  
Fractional Cable Equation
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