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 Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 8, Pages 1319–1331 (Mi zvmmf10078)

Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies

M. Yu. Medvedik, Yu. G. Smirnov, A. A. Tsupak

Penza State University, ul. Krasnaya 40, Penza, 440026, Russia

Abstract: The scalar problem of plane wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space $\mathbb{R}^3$ but rather everywhere except for the screen edges. The original boundary value problem for the Helmholtz equation is reduced to a system of weakly singular integral equations in the regions occupied by the bodies and on the screen surfaces. The equivalence of the integral and differential formulations is proven, and the solvability of the system in the Sobolev spaces is established. The integral equations are approximately solved by the Bubnov–Galerkin method. The convergence of the method is proved, its software implementation is described, and numerical results are presented.

Key words: scalar problem of plane wave diffraction, Helmholtz equation, method of singular integral equations in Sobolev space, Galerkin method, convergence of numerical scheme, software implementation.

DOI: https://doi.org/10.7868/S0044466914080109

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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:8, 1280–1292

Bibliographic databases:

UDC: 519.634
MSC: 78A45

Citation: M. Yu. Medvedik, Yu. G. Smirnov, A. A. Tsupak, “Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies”, Zh. Vychisl. Mat. Mat. Fiz., 54:8 (2014), 1319–1331; Comput. Math. Math. Phys., 54:8 (2014), 1280–1292

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. G. Smirnov, A. A. Tsupak, “Method of Integral Equations in the Scalar Problem of Diffraction on a System Consisting of a “soft” and a “hard” Screen and An Inhomogeneous Body”, Differ. Equ., 50:9 (2014), 1150–1160
2. Y. G. Smirnov, A. A. Tsupak, “Integrodifferential equations of the vector problem of electromagnetic wave diffraction by a system of nonintersecting screens and inhomogeneous bodies”, Adv. Math. Phys., 2015, 945965
3. S. A. Manenkov, “Two approaches to solving the problem of diffraction by a cylindrical body with a coordinate-dependent refractive index”, J. Commun. Technol. Electron., 61:11 (2016), 1237–1244
4. M. Yu. Medvedik, Yu. G. Smirnov, A. A. Tsupak, D. V. Valovik, “Vector problem of electromagnetic wave diffraction by a system of inhomogeneous volume bodies, thin screens, and wire antennas”, J. Electromagn. Waves Appl., 30:8 (2016), 1086–1100
5. Yu. G. Smirnov, A. A. Tsupak, “On the Fredholm property of the electric field equation in the vector diffraction problem for a partially screened solid”, Differ. Equ., 52:9 (2016), 1199–1208
6. Yu. G. Smirnov, M. Yu. Medvedik, A. A. Tsupak, M. A. Moskaleva, “Zadacha difraktsii akusticheskikh voln na sisteme tel, ekranov i antenn”, Matem. modelirovanie, 29:1 (2017), 109–118
7. E. H. Khalilov, “Substantiation of the collocation method for one class of systems of integral equations”, Ukr. Math. J., 69:6 (2017), 955–969
8. Yu. G. Smirnov, A. A. Tsupak, “Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens”, Appl. Anal., 96:8 (2017), 1326–1341
9. E. H. Khalilov, “Constructive method for solving a boundary value problem with impedance boundary condition for the Helmholtz equation”, Differ. Equ., 54:4 (2018), 539–550
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