Finite difference/spectral approximations for the fractional cable equation

Abstract

Abstract. The Cable equation has been one of the most fundamental equations for modeling neuronal dynamics. In this paper, we consider the numerical solution of 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. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is threefold: 1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order O( t 2−max{α,β} + t −1 N −m ), where t, N and m are respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. α and β are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims.

A PDE Approach to Space-Time Fractional Parabolic Problems

“…There are several approaches via finite differences, finite elements and spectral methods. For instance, a finite difference scheme is proposed and analyzed in [15,16] to deal with ∂ γ t and the so-called fractional cable equation. Semidiscrete finite element methods have been analyzed in [12] for (1.1) with γ ∈ (0, 1) and s = 1.…”

“…The finite difference scheme proposed in [15,16] has a consistency error O(τ 2−γ ), where τ denotes the time step. This error estimate, however, requires a rather strong regularity assumption in time which is problematic; see [17] and §3.2.…”

“…The main objective of this work is to describe and analyze a fully discrete scheme for problem (1.6). We use implicit finite differences for time discretization [15,16], and first degree tensor product finite elements for space discretization.…”

Exact solution for the fractional cable equation with nonlocal boundary conditions

“…Note that when δ is a positive integer, (9) can be expressed in terms of the derivative of a two-parameter Mittag-Leffler function E α β ( ) := E 1 α β ( ), see [17], eq. (9.7):…”

“…In [6] the FCE on bounded space domains for general mixed Robin boundary conditions is solved analytically. Numerical methods for solving the FCE with Dirichlet boundary conditions are developed in [7][8][9]. for α < β.…”