Beichuan Deng

Beichuan Deng, Zhimin Zhang, Xuan Zhao

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order $α$ using classical Lobatto-type polynomials when $α\in (0,1)$ and generalized Jacobi functions (GJF) for arbitrary $α> 0$, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different $α$ at the superconvergence points is at least $O(N^{-2})$ better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order $α> 1$ with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be $O(N^{-(α+3)/2})$ for $α\in (0,1)$ and $O(N^{-2})$ for $α> 1$. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.

Beichuan Deng, Jiwei Zhang, Zhimin Zhang

In this paper, we study convergence and superconvergence theory of integer and fractional derivatives of the one-point and the two-point Hermite interpolations. When considering the integer-order derivative, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are $O(N^{-2})$ and $O(N^{-1.5})$, respectively, better than the global rate for the one-point and two-point interpolations. Here $N$ represents the degree of interpolation polynomial. It is proved that the $α$-th fractional derivative of $(u-u_N)$ with $k<α<k+1$, is bounded by its $(k+1)$-th derivative. Furthermore, the corresponding superconvergence points are predicted for fractional derivatives, and an eigenvalue method is proposed to calculate the superconvergence points for the Riemann-Liouville fractional derivative. In the application of the knowledge of superconvergence points to solve FDEs, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.