PRESENCE AND UNICITY OF SOLUTION OF THE SCALAR PROBLEM OF DIFFRACTION BY A VOLUMETRIC
INHOMOGENEOUS SOLID WITH A PIECE-WISE SMOOTH REFRACTIVE INDEX 

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), E-mail: mmm@pnzgu.ru 

517.968, 517.983 

10.21685/2072-3040-2018-3-2 

Background. The aim of the present paper is investigation of the direct scalar problem of plane wave scattering by a volumetric inhomogeneous solid, characterized by piece-wise smooth refractive index.
Material and methods. The considered scattering problem is considered in the semiclassical formulation; the scattering problem is reduced to a weakly singular Fredholm integral equation of the second kind.
Results. The semiclassical formulation of the scattering problem is proposed; the uniqueness theorem is proved for the scattering problem in the original problem is reduced to the Lippmann-Schwinger integral equation; equivalency between the integral equation of the second kind and the boundary value problem is proved.
Conclusions. The obtained results on existence of a unique solution to the problem and its continuity obtained in the present article can be used for theoretical investigation of inverse problems of diffraction by compound volumetric obstacles.

diffraction problem, quasi-classical solutions, integral equations, existence and uniqueness of a solution 

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