EXISTENCE AND UNICITY OF THE SOLUTION OF THE DIFFRACTION PROBLEM FOR AN ELECTROMAGNETIC WAVE ON A SYSTEM OF NON-INTERSECTING BODIES AND SCREENS 

Valovik Dmitriy Viktorovich, Doctor of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), dvalovik@mail.ru
Medvedik Mikhail Yur'evich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street,  Penza, Russia), _medv@mail.ru
Smirnov Yuriy Gennad'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), mmm@pnzgu.ru
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 theoretically study the vector problem of electromagnetic wave scattering by an obstacle of complex shape consisting of several solid bodies and infinitely thin absolutely conducting screens.
Material and methods. The problem is considered in the quasiclassical statement (solution is sought in the classical sense everywhere except for the screen edge); to prove the theorem of uniqueness of the solution to the boundary value problem, the authors used classical integral formulas generalized for the elements of the Sobolev spaces; to prove existence and continuity of the solution, the researchers applied the theory of elliptic pseudodifferential operators over bounded manifolds.
Results. The quasiclassical statement of the electromagnrtc wave diffraction problem has been suggested; the theorem of uniqueness of the quasi-classical solution to the boundary value problem was proved; the Fredholm property of the matrix integro-differential operator was established; the theorem on continuity of solutions to the integro-differential equations was proved.
Conclusions. The obtained results can be used in the study of more complicated diffraction problems as well as for validation of numerical methods for their approximate solution.

diffraction problem, quasi-classical solutions, uniqueness theorem, Sobolev spaces, Fredholm operator, elliptic PsDO.

1. Maksimova M. A., Medvedik M. Yu., Smirnov Yu. G., Tsupak A. A. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2014, no. 3 (31), pp. 114–133.
2. Medvedik M. Yu., Smirnov Yu. G., Tsupak A. A. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki [Journal of calculus mathematics and mathematical physics]. 2014, vol. 54, no. 8, pp. 1319–1331.
3. Samokhin A. B. Integral'nye uravneniya i iteratsionnye metody v elektromagnitnom rasseyanii [Integral equations and iteration methods in magnetic scattering]. Moscow: Radio i svyaz', 1998, 160 p.
4. Il'inskiy A. S., Smirnov Yu. G. Difraktsiya elektromagnitnykh voln na provodyashchikh tonkikh ekranakh [Diffraction of electromagnetic waves on conductive thin screens]. Moscow: Radiotekhnika, 1996, 176 p.
5. Il'inskiy A. S., Kravtsov V. V., Sveshnikov A. G. Matematicheskie modeli elektrodinamiki [Mathematical models of electrodynamics]. Moscow: Vysshaya shkola, 1991, 224 p.
6. Costabel M., Darrigrand E., Kone E. J. Comput. Appl. Math. 2010, no. 234, pp. 1817–1825.
7. Valovik D. V., Smirnov Yu. G. Differentsial'nye uravneniya [Differential equations]. 2012, vol.48,no.4,pp.509–515.