Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have min{2 - alpha, 2 - beta}-order convergence in time and spectral accuracy in space for smooth solutions, where alpha, beta are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow.