Finite Difference/Spectral Approximations For The Nonlinear Fractional Cable Equation

The nonlinear fractional cable equation is one of the most basic equations of neu-ron dynamics.This paper studies the numerical solution of the nonlinear fractional Cable equation,and considers the anomalous diffusion in ion motion in the neuron system.First,in the time direction,we obtain the finite difference semi-discrete format after the backward Euler difference format is approximately discrete.In the spatial direction,on the basis of semi-discrete problems,using Legendre poly-nomials as a base and Gauss-Lobatto-Legendre points as integration points,we obtained a fully discrete format by spectral Galerkin method discrete.Secondly,we analyzed the convergence of the format,and proved that this format is uncon-ditionally stable.Next,the error was analyzed,and detailed verification was per-formed.The error of the fully discrete numerical solution accurately converged to O(?tmin(2-?,2-?)+?t-?N1-m),where ?t,N,m are time steps,polynomial degree and space dimension,?,? is two fractions related to fractional derivative between 0 to 1.Finally,we prove the theory through numerical experiments.