Spectral methods for the time fractional diffusion–wave equation in a semi-infinite channel

In this paper, we consider the numerical approximation of the time fractional diffusion–wave equation in a semi-infinite channel. The time fractional derivative is described in Caputo sense with order γγ   (1<γ<2)(1<γ<2). A fully discrete spectral scheme based on a finite difference method in the time direction and a Laguerre–Legendre spectral method in the space direction is proposed. We also propose an alternating direction implicit (ADI) spectral scheme in order to reduce the amount of computation. The stability and convergence of both schemes are rigorously established. Numerical results are presented to support our theoretical analysis.