An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of $H$-polarized electromagnetic waves by a screen with a curved boundary, are constructed. This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of $20 - 30$ points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.