Superconvergence Points of Integer and Fractional Derivatives of Special Hermite Interpolations and Its Applications in Solving FDEs

Abstract: 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 $\alpha$-th fractional derivative of $(u-u_N)$ with $k<\alpha<k+1$, 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.