Application of GRID technologies for solving a nonlinear volumetric singular integral equation to determine the effective permittivity of nanomaterials

The work is devoted to the study of the problem of determining the effective dielectric constant of samples of nanomaterials of arbitrary geometric shape placed in a rectangular waveguide with ideally conductive walls. The problem is reduced to solving a nonlinear volumetric singular integral equation. The study of the integral equation is based on the results of the study of the corresponding boundary value problem and the equivalence theorem of the boundary value problem and the integral equation. The theorem on the existence and uniqueness of solutions in the $L_2$ integral equation, the convergence of the Galerkin numerical method are proved, and results on the smoothness of solutions are obtained. A parallel computational algorithm and a procedure for using GRID technologies to solve the problem are proposed.