Numerical Solution of the Time Fractional Cable Equation

Abstract

The time fractional diffusion equation has attracted the attention of many researchers in the last years due to its many applications in different domains. In this article we are concerned with one of these models, the time fractional cable equation:

$$\begin{aligned} \frac{\partial ^2 V(x,t)}{\partial x^2} -\frac{\partial ^{\alpha } V(x,t)}{\partial x^{\alpha }}-V(x,t)=0, \end{aligned}$$

(1)

which describes the spatial and temporal dependence of transmembrane potential V(x, t) along the axial direction of a cylindrical nerve cell segment. Here $\frac{\partial ^{\alpha } V(x,t)}{\partial x^{\alpha }}$ is a Caputo-type derivative, with $0<\alpha <1$. We use a numerical scheme to solve Eq. (1), which is based on the L1-method and on a finite-difference scheme for the time and space discretization, respectively. In order to deal with the singularity at $t=0$ we use non-uniform meshes. Numerical examples are presented which illustrate the efficiency of the method.