The Grunwald formula is the traditional method to deal with Riemann–Liouville fractional derivative, while its convergence is only [Formula: see text]. In this paper, a high-order polynomial approximation is presented for the Riemann–Liouville fractional derivative. The quadratic polynomial functions and their fractional derivatives with explicit expressions are constructed to approximate the fractional derivative instead of Grunwald formula or shifted Grunwald formula. We proved that this technique has convergence of [Formula: see text]. Based on the MLS approximation and the high-order polynomial approximation and center difference method, a meshless analysis is proposed for the two-dimensional two-sided space-fractional wave equations (SFWE). The SFWE is found to be very adequate in describing anomalous transport and dispersion phenomena. In the meshless method, the trial function for the SFWE is constructed by the MLS approximation and the Riemann–Liouville fractional derivative is approximated by the high-order polynomial approximation, and the essential boundary conditions can be directly and easily imposed on as finite element method. This technique avoids singular integral, and has high accuracy and efficiency. Numerical results demonstrate that this method is highly accurate and computationally efficient for space-fractional wave equations.
Analysis of the L1 scheme for fractional wave equations with nonsmooth data
