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 $O(\triangle t^{2-\max \{\alpha ,\beta \}}+ \triangle t^{-1}N^{-m})$ , where $\triangle t,N$ and $m$ are respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. $\alpha$ and $\beta$ 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.

In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than H1 ). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.