Generalization of the method of extended boundary conditions

The electrostatic approximation in the problem of diffraction of a plane wave by a group of coaxial small scatterers

“…The potential in the interior of each particle has the form (8) According to the MNFM, it is necessary to require that the following conditions be fulfilled [2][3][4][5][6][7]: …”

“…When the problem is solved with the help of the standard MNFM, auxiliary surfaces are constructed in spherical coordinates. Namely, if the original surface of a particle is specified in spherical coordinates in the form r = r(θ), the aux iliary surfaces are determined by the equations [2][3][4][5][6][7] (11)…”

“…The requirement that this condition be fulfilled on a certain closed surface inside a particle enables one to reduce the boundary value problem to the solution of a Fredholm integral equation of the first kind with a smooth kernel. In studies [2][3][4][5][6][7][8], it is shown that the fastest and most stable algorithms can be developed when the auxiliary surface is constructed with the help of analytical deformation of a scatterer's boundary. When a problem of diffraction by compact obstacles (in the 3D case) or cylindrical bodies (the 2D case) is solved, the auxiliary surface is constructed in spherical or polar coordinates.…”

“…Earlier, the MNFM was applied to a wide class of diffraction problems including problems of scattering by single bodies [2][3][4][5], periodic arrays [6,7], a wavy medium interface [8], etc. The MNFM is based on the so called null field condition satisfied for any point inside a body.…”