| 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−uN), 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.
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