Optimal quadrature formula for approximate solution of a singular integral equation with the Hilbert kernel in the space $L_2^{(3)}(0, 2\pi)$

Abstract

In this article, an optimal quadrature formula for the approximate calculation of the solution of a Fredholm-type singular integral equation of the second kind with the Hilbert kernel in the space $L_2^{ (3) } (0, 2\pi) $ is constructed. In the proposed quadrature formula, the Sard problem arises, where the interval $[0, 2\pi]$ is evenly divided into any $N$ points, and this problem is solved using the Sobolev method. For this purpose, an analytical representation of the optimal coefficients of the quadrature formula is obtained by minimizing the norm of the error functional in the conjugate space $L_2^{ (3) ^*} (0, 2\pi) $.