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A fast‐high order compact difference method for the fractional cable equation
Author(s) -
Liu Zhengguang,
Cheng Aijie,
Li Xiaoli
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22286
Subject(s) - mathematics , discretization , matrix difference equation , finite difference method , toeplitz matrix , norm (philosophy) , finite difference , diffusion equation , mathematical analysis , cable theory , rate of convergence , finite element method , compact finite difference , partial differential equation , riccati equation , cable harness , computer science , pure mathematics , economy , cable gland , law , service (business) , computer network , telecommunications , political science , thermodynamics , physics , economics , channel (broadcasting)
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 ( τ min ( 1 + γ 1 , 1 + γ 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 ( M N 2 ) required by traditional methods to O ( M N log 2 N ) without using any lossy compression, where N = τ − 1and τ is the size of time step, M = h − 1and 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.