Galerkin spectral method for nonlinear time fractional Cable equation with smooth and nonsmooth solutions

Abstract

In this work, we study the numerical solutions of the time fractional Cable equations with nonlinear term, where the fractional derivatives are described in Riemann–Liouville sense. An explicit scheme is constructed based upon finite difference method in time and Legendre spectral method in space. Stability and convergence of scheme are proved rigorously. Moreover, an improved algorithm for the problem with nonsmooth solutions is proposed by adding correction terms to the approximations of first-order derivative, fractional derivatives and nonlinear term. Numerical examples are given to support theoretical analysis.