On the numerical solution to a weakly singular integral equation of the second kind by the method of orthogonal polynomials

It is considered a singular integral equation with a logarithmic singularity. Such equations are used in the mathematical model of electromagnetic wave scattering. Three computational schemes are constructed for the numerical analysis of its solutions from different Muskhelishvili functional classes. They are based on the representation of a part of the determined function as a linear combination of Chebyshev polynomials of the first kind. After minor transformations and application of the known spectral relations for the singular integral, simple analytical expressions for the singular component of the equation are obtained. The solution is expanded in the basis of Chebyshev polynomials. The expansion coefficients are calculated as the solution of the corresponding systems of linear algebraic equations. The results of numerical experiments show that on a grid of 15–20 nodes, the error of the approximation does not exceed the computational error.