Integral equations in the wave scattering problem at an irregular interface of domains

Modeling the scattering of electromagnetic waves at an interface of media with different characteristics, one encounters the conjugation problem. Using the method of boundary integral equations and the theory of generalized potentials, we prove the classical resolvability of this problem. The boundary is assumed to be irregular. This means that the plane is divided into two domains by a curve which coincides with a straight line, except for a finite part, producing the irregularity. We propose algorithms for the approximate solution of the conjugation problem based on the spline methods for the solution of integral equations. We theoretically substantiate the computational scheme, namely, we prove the convergence and estimate the convergence rate.