An approximate method for solving a characteristic singular integral equation

Many problems in science and engineering are naturally reduced to singular integral equations. Moreover, planar problems are reduced to one-dimensional singular integral equations. In the present paper, we develop an optimal algorithm for the approximate solution of one-dimensional singular integral equations with the Cauchy kernel. Here, we focus on finding the analytical form of the coefficients of the optimal quadrature formula. We apply these coefficients to an approximate solution of the Fredholm singular integral equation of the first kind. Thus, we demonstrate the possibility of solving singular integral equations with higher accuracy using the optimal quadrature formula.