Fractional differential equations arise in various applied areas of science and engineering, for example, in the “modeling of anomalous diffusive and sub-diffusive systems, [the] description of fractional random walk, and [the] unification of diffusion and wave propagation phenomena.” The numerical solution of fractional diffusion-wave equations has been getting a lot of attention from researchers in recent years. In general, it is very difficult to design efficient numerical schemes for fractional differential equations for both the spatial and temporal discretizations.

The authors of this paper construct some effective numerical schemes for the time fractional diffusion-wave equation with Neumann boundary conditions, and obtain the corresponding error estimates. The L₁ approximation is used to discretize the Caputo fractional derivative in the temporal direction, and a fourth-order compact difference scheme is used for the spatial derivative. The proposed difference scheme is unconditionally stable and the order of convergence in the maximum norm is O(τ^3-α + h⁴), where τ and h denote the grid size in the temporal and spatial directions, respectively.

The authors use the Crank-Nicolson scheme to solve the equation, and derive the error estimates. For the 2D case, they derive a compact alternating direction implicit (ADI) scheme, and show that the global order of convergence is O(τ^3-α + h_1⁴ + h_2⁴) in the standard H¹-norm, where τ, h₁, and h₂ are respectively the step sizes in the temporal, x, and y directions. Several numerical examples are presented to validate the theoretical results.

This interesting paper provides efficient numerical schemes for fractional differential equations and the corresponding stability analysis and error estimates.