On the Fredholm Property of a Hypersingular Integral Operatorin a Problem of Electromagnetic Wave Scattering on a Slab Coveredwith Graphene

The paper focuses on a boundary value problem for a system of two homogeneous Helmholtz equations with mixed boundary conditions arising in the theory of electromagnetic wave scattering on structures covered with two-dimensional materials. The problem is reduced to a boundary integral equation on the entire real line involving a hypersingular integral operator. Introducing new variables and unknown function, one passes to an integral equation on an interval. To find an approximate solution of the resulting equation, a collocation method using Fourier–Chebyshev series to represent the solution is suggested; hypersingular integrals are calculated analytically. In the main part of the paper, we prove the Fredholm property of the hypersingular operator involved in the integral equation under consideration.