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.

Differential-algebraic equation (DAE) is widely used in engineering domains, such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory, owing to their ability to effectively characterize dynamic variations and inherent constraints. In recent years, physics-informed neural networks (PINNs) have manifested remarkable advantages in solving both the forward and inverse problems of DAE by integrating physical prior knowledge into neural network models. Presently, PINNs-based approaches still encounter challenges of inadequate solution accuracy and limited generalization performance when dealing with DAE involving discrete events. This paper presented a physics-informed neural network that integrates the Legendre-Gauss-Lobatto (LGL) collocation method from spectral methods to solve the aforementioned DAE with discrete event. To further augment the accuracy and continuity of the solution, the model employed a time-domain decomposition strategy to construct the network architecture, thereby enabling high-precision continuous-time prediction of DAE. Numerical examples illustrated that the LGL-PINN can attain high-precision solutions of DAE. In comparison with the PINNs, the error between the predicted solution and the exact solution of the LGL-PINN was substantially reduced, with the accuracy improved by one to two orders of magnitude. Therefore, the proposed solution model demonstrated excellent computational accuracy for solving DAE problems involving discrete event.