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Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order α ∈ (1 , 2) is proposed. This method is different from the known second-order methods, which is called the L 2- 1 σ Crank-Nicolson method in this paper. Using the L 2- 1 σ Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L 2- 1 σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L 2 -norm and H 1 -norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all α ∈ (1 , 2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L 2- 1 σ Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes. [ABSTRACT FROM AUTHOR]

In a recent paper, Ren and Liu proposed and analyzed a high-order compact finite difference method for a class of fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions. In this paper, we point out some deficiencies and errors found in that paper and make the corresponding revisions. [ABSTRACT FROM AUTHOR]