An optimal approximate solution of the I kind Fredholm singular integral equations

In this work, we study the problem of profile and lattice profile in aerodynamics. The first kind of Fredholm singular integral equation solves this problem. The Cauchy-type singular integrals are solutions to the first kind of Fredholm singular integral equation. However, the antiderivative function of these singular integrals can only be found in some cases. To overcome this, we utilize the Sobolev method to create an optimal quadrature formula for Cauchy-type singular integrals. By doing so, we can approximately solve the Fredholm integral equation of type I with higher accuracy. We compare the exact and approximate solutions of the first kind Fredholm singular integral equation, utilizing both the Sobolev method and another approach.