ON FREDHOLM PROPERTY OF AN INTEGRO-DIFFERENTIAL OPERATOR IN THE PROBLEM OF                                        ELECTROMAGNETIC WAVE DIFFRACTION ON A VOLUMETRIC BODY, PARTIALLY                             SCREENED BY A SYSTEM OF FLAT SCREENS

Tsupak Aleksey Aleksandrovich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), altsupak@yandex.ru

517.968, 517.983.37, 517.958:535.4

Background. The aim of this work is to study a new vector problem of electromagnetic wave scattering on a partially shielded volumetric inhomogeneous anisotropic body.
Material and methods. The problem is considered in the quasiclassical formulation; the original boundary value problem is reduced to a system of integro-differential equations; the properties of the system are studied using pseudodifferential calculus in Sobolev spaces on manifolds with a boundary.
Results. The quasiclassical formulation of the diffraction problem is proposed; the boundary value problem for Maxwell’s equations is reduced to a system of integrodifferential equations; the operator of this system is treated as a pseudodifferential operator (ψDO) in Sobolev spaces on manifolds with a boundary; the quadratic form of the matrix ψDO is studied and is shown to be coercive; the Fredholm property of the ψDO is proved.
Conclusions. The matrix ψDO is proved to be a Fredholm operator of zero index; this results can be used for further theoretical study of the diffraction problem as well as for validation of numerical methods.

vector diffraction problem, integro-differential equations, Sobolev spaces, pseudodifferential operators, coercive quadratic form. 

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