Energy estimates for two-dimensional space-Riesz fractional wave equation

The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic viscoelasticity. In this paper, we first develop the energy method to estimate the one-dimensional space-Riesz fractional wave equation. For two-dimensional cases with the variable coefficients, the discretized matrices are proved to be commutative, which ensures to carry out of the priori error estimates. The unconditional stability and convergence with the global truncation error $\mathcal{O}(\tau^2+h^2)$ are theoretically proved and numerically verified. In particulary, the framework of the priori error estimates and convergence analysis are still valid for the compact finite difference scheme and the nonlocal wave equation.