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 $ \Phi(\tau) $ are attained. It is further demonstrated that the superconvergence phenomenon is uniformly discerned whenever the singular point coincides with the zeros of $ \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.
Citation: Wenxuan Zhao, Dongxin Guo, Jin Li, Qingli Zhao. Cubic spline rule to compute hypersingular integral on a circle[J]. Electronic Research Archive, 2026, 34(5): 3008-3023. doi: 10.3934/era.2026136
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 $ \Phi(\tau) $ are attained. It is further demonstrated that the superconvergence phenomenon is uniformly discerned whenever the singular point coincides with the zeros of $ \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.
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Wenxuan Zhao, Dongxin Guo, Jin Li, Qingli Zhao. Cubic spline rule to compute hypersingular integral on a circle[J]. Electronic Research Archive, 2026, 34(5): 3008-3023. doi: 10.3934/era.2026136
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