Cubic spline quadrature rule to calculate supersingular integral on interval

The cubic spline quadrature rule for the calculation of supersingular integral (also called “third order hypersingular integral”) is discussed. The superconvergence phenomenon exists at the midpoint of subinterval and the superconvergence point is the zero point of the special function. When τ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =0$$\end{document}, the order of convergence at the superconvergence point is higher than that at the non-superconvergence point. The superconvergence theory of the cubic spline quadrature function for the supersingular integral can be proved by hermite quadrature formula. Finally, examples are given to illustrate the effectiveness of the proposed method.