Problems of electromagnetic wave scattering on 3D dielectric structures in the presence of bounded perfectly conducting surfaces are reduced to a system of singular integral equations. We study this system mathematically and suggest a numerical solution method.
Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.
We consider the problem of leaky waves in an inhomogeneous waveguide structure coveredwith a layer of graphene, which is reduced to a boundary value problem for the longitudinalcomponents of the electromagnetic field in Sobolev spaces. A variational statement of the problemis used to determine the solution. The variational problem is reduced to the study of an operatorfunction. The properties of the operator function necessary for the analysis of its spectralproperties are investigated. Theorems on the discreteness of the spectrum and on the distributionof the characteristic numbers of the operator function on the complex plane are proved.