@article{Zhao2026, 
author = {Wenxuan Zhao and Dongxin Guo and Jin Li and Qingli Zhao},
title = {Cubic spline rule to compute hypersingular integral on a circle},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {5},
pages = {3008-3023},
keywords = {hypersingular integral, cubic spline interpolation, Hadamard finite-part integral, superconvergence, Clausen functions},
url = {https://www.sciopen.com/article/10.3934/era.2026136},
doi = {10.3934/era.2026136},
abstract = {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    Φ  (  τ  ) are attained. It is further demonstrated that the superconvergence phenomenon is uniformly discerned whenever the singular point coincides with the zeros of    Φ  (  τ  ), 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.}
}