A fast-high order compact difference method for the fractional cable equation

The Cable equation is one of the most fundamental equations for modeling neuronal dynamics. In this article, we consider a high order compact finite difference numerical solution for the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. The resulting finite difference scheme is unconditionally stable and converges with the convergence order of O(tau(min(1+gamma 1,1+gamma 2)) + h(4)) in maximum norm, 1-norm and 2-norm. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from O(MN2) required by traditional methods to O(MN log(2) N) without using any lossy compression, where N = tau(-1) and tau is the size of time step, M = h(-1) and h is the size of space step. Moreover, we give a compact finite difference scheme and consider its stability analysis for two-dimensional fractional Cable equation. The applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.