Cubic spline rule to compute hypersingular integral on a circle

A novel approach for the high-precision evaluation of hypersingular integrals on a circle by the spline approximation of the periodic density function is presented. A cubic spline interpolation function with periodic boundary conditions, enforced through a cyclic tridiagonal system, is constructed through the uniform partitioning of the periodic interval. Through the analytical properties of the Clausen functions, an explicit expression for the integral is derived, and a rigorous error analysis is conducted. Theoretical results demonstrate that a convergence rate of $O(h^3)$ at non-superconvergent points and $O(h^4)$ superconvergence at the zeros of the special function $\mathrm{\Phi }(\tau )$ are attained. It is further demonstrated that the superconvergence phenomenon is uniformly discerned whenever the singular point coincides with the zeros of $\mathrm{\Phi }(\tau )$, regardless of the singular point's relative position within the mesh. Finally, a numerical example is presented for illustrating the effectiveness of the proposed method. The computed errors across diverse mesh sizes and singular point locations are in remarkable agreement with theoretical predictions.