Superconvergence points of integer and fractional derivatives of special Hermite interpolations and its applications in solving FDEs

¹ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
² Beijing Computational Science Research Center, 100193, Beijing, PR China

^* Corresponding author: zmzhang@csrc.ac.cn

Received: 18 July 2018
Accepted: 21 February 2019

Abstract

In this paper, we study the theory of convergence and superconvergence for integer and fractional derivatives of the one-point and two-point Hermite interpolations. When considering the integer-order derivatives, 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) better than the global rates for the one-point and two-point interpolations, respectively. Here N represents the degree of the interpolation polynomial. It is proved that the αth 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 derivatives. 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.