Superconvergence Points of Integer and Fractional Derivatives of Special Hermite Interpolations and Its Applications in Solving FDEs
For researchers in numerical analysis and fractional differential equations, this work provides theoretical foundations and practical improvements for solving FDEs, though it is incremental as it extends known superconvergence concepts to fractional derivatives.
This paper proves superconvergence points for integer and fractional derivatives of Hermite interpolations, achieving convergence rates up to O(N^{-2}) for integer derivatives, and proposes a modified collocation method for fractional differential equations that significantly improves accuracy over traditional methods.
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 $α$-th fractional derivative of $(u-u_N)$ 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 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.