An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space

This paper addresses the problem of constructing optimal quadrature formulas for the numerical evaluation of Hadamard-type hypersingular integrals with hihg oscillation. Such integrals frequently arise in the analysis of boundary-value problems for partial differential equations, particularly in hyperbolic and wave-type models, as well as in various applied fields such as elasticity theory, fluid dynamics, and electromagnetic scattering. Our primary focus is on deriving the analytical expressions for the coefficients of an optimal quadrature formula tailored to the structure of the singular and oscillatory kernel. The goal is to achieve minimal error in a rigorous sense by constructing formulas that are optimal in the sense of Sard, meaning that they minimize the worst-case error over a specified function class. To this end, we develop an optimal quadrature formula of the form (2) in the Sobolev space, applying the Sobolev method for constructing optimal quadrature formulas. This method leverages variational principles and operator-theoretic tools to systematically minimize the norm of the error functional. The resulting formulas are well-adapted to handle the challenging combination of hypersingularity and oscillation, and they offer both theoretical optimality and practical computational advantages.