Quadrature Formulas with Prescribed Nodes for Singular Integrals

Abstract

In applied problems, one often needs to derive quadrature formulas in which some of the nodes are given in advance. For example, a boundary value problem for a second-order differential equation on the interval [a, b] with boundary conditions at the endpoints a and b naturally requires the use of quadrature formulas containing the nodes a and b (see [1]). The consideration of such quadrature formulas had been initiated by Markov, whose results were later generalized by Krylov, but he had derived formulas only for regular integrals. The comprehensive development of the theory of singular integrals and singular integral equations is due to the fact that the solutions of many important problems can be expressed via integrals with Cauchy kernel. It is of interest to develop efficient approximate methods for computing such integrals. Investigations in this direction were continued by Ivanov, Lifanov, Sheshko, Sanikidze, and others. In the present paper, for the singular integral