Two New Implicit Numerical Methods for the 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.