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

TL;DR

This paper investigates superconvergence points in integer and fractional Hermite interpolations, providing theoretical error decay rates, locating superconvergence points, and applying these findings to improve fractional differential equation solutions.

Contribution

It introduces new superconvergence results for integer and fractional derivatives of Hermite interpolations and develops an eigenvalue method to compute superconvergence points for fractional derivatives.

Findings

Superconvergence points yield higher convergence rates than global rates.

The eigenvalue method effectively calculates superconvergence points for Riemann-Liouville derivatives.

Modified collocation methods significantly improve FDE solutions.

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.