Approximate solving of the third boundary value problems for Helmholtz equations in the plane with parallel cuts
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 3, pp. 254-267 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the paper, the method of approximate solution of boundary integral equations of the original problem is proposed. The systems of boundary integral equations of the problem are obtained by the method of parametric representation of integral transforms. The convergence of approximate solutions to the exact solution of the original problem is guaranteed by the propositions proved in the paper. Also, the rate of convergence of approximate solutions to exact solutions is found.
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V. D. Dushkin. Approximate solving of the third boundary value problems for Helmholtz equations in the plane with parallel cuts. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 3, pp. 254-267. http://geodesic.mathdoc.fr/item/JMAG_2017_13_3_a2/

[1] A.S. Il'insky, A.Ja. Slepjan, G.Ja. Slepjan, Propagation, Diffraction and Dissipation of Electromagnetic Waves, Electromagnetic Waves, 36, The IEE and Peter Peregrinous Ltd., London, UK, 1993 | MR

[2] N.I. Akhiezer, Lectures on Integral Transforms, Translations of Mathematical Monographs, 70, Amer. Math. Soc., Providence, RI, 1988 | DOI | MR | Zbl

[3] Yu.V. Gandel, “Parametric Representations of Integral and Psevdodifferential Operators in Diffraction Problems”, Proc. 10th Int. Conf. on Math. Methods in Electromagnetic Theory (Dnepropetrovsk, Ukraine, Sept. 14–17, 2004), 57–62

[4] Yu.V. Gandel', “Boundary-Value Problems for the Helmholtz Equation and their Discrete Mathematical Models”, J. Math. Sci., 171:1 (1990), 74–88

[5] Yu.V. Gandel, V.D. Dushkin, “The Method of Parametric Representations of Integral and Pseudo-differential Operators in Diffraction Problems on Electrodynamic Structures”, Proceedings of the International Conference Days on Diffraction DD (2012, 28 May–1 June, 2012), 76–81

[6] Yu.V. Gandel, V.D. Dushkin, Mathematical Models of Two-Dimensional Diffraction Problems: Singular Integral Equations and Numerical Methods of Discrete Singularities Method, Kharkiv, 2012 (Russian)

[7] S.M. Belotserkovsky, I.K. Lifanov, Method of Discrete Vortices, CRC Press, New York, 1993 | MR

[8] I.K. Lifanov, Singular Integral Equations and Discrete Vortices, VSP, Utrecht–Tokyo, 1996 | MR | Zbl

[9] Yu.V. Gandel', T.S. Polyanskaya, “Justification of a Numerical Method for Solving Systems of Singular Integral Equations in Diffraction Grating Problems”, Differ. Equ., 39:9 (2003), 1295–1307 | DOI | MR | Zbl

[10] Yu. V. Gandel', S. V. Eremenko, T. S. Polyanskaya, Mathematical Problems in the Method of Discrete Currents. Justification of the Numerical Method of Discrete Singularities of Solutions of Two-Dimensional Problems of Diffraction of Electromagnetic Waves. Educational aid, v. II, Kharkov State University, Kharkov, 1992 (Russian)

[11] V.D. Dushkin, “The Justification of Numerical Solution of Boundary Integral Equations of Wave Scattering Problems on Impedance Lattice”, Visn. Kharkiv. Nats. Univ., Mat. Prikl. Mat. Mekh., 1120:69 (2014), 20–28 | Zbl

[12] Yu.V. Gandel', V.D. Dushkin, “The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice”, Am. J. App. Mathematics and Statistics, 2:6 (2014), 369–375 | DOI

[13] Yu.V. Gandel', V.D. Dushkin, “The Boundary Integral Equations of the Third Boundary-Value Problem for the Helmholtz Equation in the $R_{+}^{2} $ with Plane-Parallel Slits”, Dopov. Nats. Akad. Nauk Ukr., 8 (2014), 14–19 (Russian) | DOI | MR | Zbl

[14] Yu.V. Gandel', V.F. Kravchenko, V.I. Pustovoit, “Scattering of Electromagnetic Waves by a Thin Superconducting Band”, Dokl. Math., 54:3 (1996), 959–961 | MR | Zbl

[15] Yu.V. Gandel', Introduction to Methods of Evaluation of Singular and Hypersingular Integrals, Izd. Kharkov. Nats. Univ., Kharkov, 2002 (Russian)

[16] S.S. Kutateladze, Fundamentals of Functional Analysis, Kluwer Academic Publishers Group, Dordrecht, Netherlands, 1996 | MR

[17] I.P. Natanson, Constructive Function Theory, v. 1, Frederic Ungar Puplishing Co., New York, 1964 | MR | Zbl

[18] B.G. Gabdulkhaev, The Optimal Approximation of Solutions of Linear Problems, Kazan. Univ. Publishing, Kazan, 1980 (Russian) | MR

[19] Yu.V. Gandel', G.L. Sidel'nikov, “The Method of Integral Equations in the Third Boundary-Value Problem of Diffraction on a Bounded Grating Over a Flat Screen”, Differ. Equ., 35:9 (1999), 1169–1175 | MR | Zbl

[20] Yu.V. Gandel', V.D. Dushkin, “Mathematical Model of Polarized Wave Scattering on Impedance Strips located on Screened Dielectric Layer”, Mat. Metodi Fiz.-Mekh. Polya, 57:1 (2014), 125–132 | MR | Zbl

[21] Yu.V. Gandel', V.D. Dushkin, “Mathematical Model of Polarized Wave Scattering on Impedance Strips located on Screened Dielectric Layer”, J. Math. Sci., 212:2 (2016), 156–166 | DOI | MR

[22] V.D. Dushkin, “Mathematical Models of Plane Wave Scattering on Multilayer Impedance Structures”, Visn. Lviv. Univ. Prikl. Mat. Inform., 2013, no. 20, 69–76