Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order $\gamma \in (1,2)$ γ ∈ ( 1 , 2 ) . We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the $L_{1}$ L 1 approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.