Optimal quadrature formula in the space of differentiable functions

Abstract

Numerical integration in Sobolev spaces often relies on optimal quadrature formulas in the sense of Sard to minimize the calculation error. While many standard formulas exist, constructing optimal rules that incorporate a specific combination of data—such as function values at nodes, first-order derivatives at interval boundaries, and third-order derivatives at nodes—presents a complex mathematical challenge, as previous studies lack an explicit framework for deriving exact analytical coefficients. The primary objective of this study is to construct optimal quadrature formulas in the sense of Sard by minimizing the error functional with these mixed derivative data. The study employs a variational method to construct the formula: an analytical expression for the norm of the error functional is first derived, the Lagrange multiplier method is applied to generate a Wiener–Hopf type system of linear equations, and the Sobolev method is utilized to extract the exact analytical form of the optimal coefficients. Through the applied approach, precise analytical expressions for optimal quadrature coefficients are successfully obtained, and the coefficients for the spaces ????(0, 1) and ????(0, 1) are fully calculated as an exact application. The explicit solution confirms that the newly found coefficients strictly minimize the error functional, significantly enhancing the mathematical toolkit for numerical integration in Sobolev spaces.