On the smoothness of solutions of electric field volume singular integro-differential equation

Background. The goal of this paper is to study the smoothness of solutions of a volume singular integro-differential equation of electric field arising in the diffraction problem of an electromagnetic wave on a local bounded inhomogeneous dielectric body. Material and methods. The main method of the research was the method of pseudodifferential operators acting in Sobolev spaces. The theory of elliptic boundary value problems and transmission problems was also used. Results. It is proved that if the data of the problem is smooth then the square-integrable solution of the equation will be continuous up to the boundary and smooth inside and outside the body. Conclusions. Smoothness properties of solutions of the electric field volume singular integro-differential equation allow to investigate the equivalence of the boundary value problem and the equation.