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

Deng, Beichuan; Zhang, Jiwei; Zhang, Zhimin

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 1061-1082.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

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 $O\left(N^{-2}\right)$and $O\left(N^{-1.5}\right)$ 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 $\left(u-u_N\right)$, 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 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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