Consecutive optimization of the weighted quadrature formulas with derivative

Abstract

This article addresses the derivation and analysis of a weighted optimal quadrature formula in the Hilbert space $W_2^{(2,1)}(0,1)$, where functions $\varphi$ with prescribed properties reside. The quadrature formula is expressed as a linear combination of function values and its first-order derivative at equidistant nodes in the interval $[0,1]$. The coefficients are determined by minimizing the norm of the error functional in the dual space $W_2^{(2,1)*}(0,1)$. The error functional is defined as the difference between the integral of a function over the interval and the quadrature approximation. The key results include explicit expressions for the coefficients and the norm of the error functional.

The optimization problem is formulated and solved, leading to a system of linear equations for the coefficients. Analytical solutions of the system are obtained via the Sobolev method, which provides an explicit expression for the optimal coefficients. The convergence with the exact solution of the integral equations is analyzed via numerical experiments.