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In this paper, we study convergence and superconvergence theory of integerand fractional derivatives of the one-point and the two-point Hermiteinterpolations. When considering the integer-order derivative, exponentialdecay of the error is proved, and superconvergence points are located, at whichthe convergence rates are $O(N^{-2})$ and $O(N^{-1.5})$, respectively, betterthan 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 boundedby its $(k+1)$-th derivative. Furthermore, the corresponding superconvergencepoints are predicted for fractional derivatives, and an eigenvalue method isproposed to calculate the superconvergence points for the Riemann-Liouvillefractional derivative. In the application of the knowledge of superconvergencepoints to solve FDEs, we discover that a modified collocation method makesnumerical solutions much more accurate than the traditional collocation method.

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