Method of the Singularly Perturbed Differential Equations in Internal Problems of Diffraction in Polygonal Domains

The problem of finding the eigenfunctions of the Storm-Liouville problem for the Helmholtz equation with the boundary conditions of the first type in the domains without a special symmetry is considered. Eigenfunctions are built as a product of three functions, two of which satisfy the boundary conditions and the third is found from the condition that the eigenfunction must satisfy the Helmholtz equation. For this third factor a differential equation is obtained that is singularly perturbed near to the boundary. For this equation the methods of construction of regular solutions, and also solutions of boundary layer´s type are specified in an internal vicinity of borders of domains