Wayne State University Dissertations

WSU Access

January 2018

Dissertation

Ph.D.

Mathematics

Zhimin Zhang

Abstract

In this work, we study the theory of convergence and superconvergence for integer and fractional derivatives of the polynomial and GJF fractional interpolations. When considering the integer-order derivatives of Hermite interpolation, 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.

When considering fractional derivatives of polynomial interpolations, It is proved that the $\alpha$-th fractional derivative of $(u-u_N)$, with $k<\alpha Furthermore, the interpolation is generalized to the Riesz derivative of order$\alpha > 1$with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be$O(N^{-(\alpha+3)/2})$for$\alpha \in (0,1)$and$O(N^{-2})$for$\alpha > 1\$.

Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points. In the other application, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.

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