Method of Orthogonal Polynomials for an Approximate Solution of Singular Integro-Differential Equations As Applied to Two-Dimensional Diffraction Problems

We consider a mathematical model of scattering of H -polarized electromagnetic waves by a screen with a curvilinear boundary based on a singular integro-differential equation with a Cauchy kernel and a logarithmic singularity. The integrands contain both the unknown function and its first derivative. For the numerical analysis of this model, two computational schemes are constructed based on the representation of the unknown function in the form of a linear combination of orthogonal Chebyshev polynomials and spectral relations, which permit one to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the basis of Chebyshev polynomials are calculated as a solution of the corresponding system of linear algebraic equations. The results of numerical experiments show that the error in the approximate solution on a grid of 20–30 nodes does not exceed the roundoff error.