The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

Maurya V.N.^1, , Gandel Yu. V.² and Dushkin V.D.³

¹Department of Mathematics and Statistics, School of Science & Technology, The University of Fiji, Fiji

²Department of Mathematical Physics and Computational Mathematics, Karazin Kharkiv National University, Kharkiv, Ukraine

³Department of Fundamental Science, National Academy of NGU, Kharkiv, Ukraine

Cite this paper:
Maurya V.N., Gandel Yu. V. and Dushkin V.D.. The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice. American Journal of Applied Mathematics and Statistics. 2014; 2(6A):13-19. doi: 10.12691/ajams-2-6A-3

Abstract

In this article the method for numerical solution of boundary integral equations of the original problem is proposed. This method is one of the modifications of Nystrom-type methods; particularly the method of discrete vortices. The convergence of the numerical solutions to the exact solution of the problem is guaranteed by propositions proved in this article. Also, the rate of convergence of the approximate solutions to the exact solution had been found.

Keywords:
singular integral equation modification of method of discrete vortices existence of approximate solution the rate of convergence of the approximate solutions

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Bulygin, V.S., Benson, T.M., Gandel, Y.V., Nosich, A.I., “Full-wave analysis and optimization of a TARA-like shield-assisted paraboloidal reflector antenna using a nystrom-type method,” IEEE Transactions on Antennas and Propagation, 61(10), 4981-4989, 2013.
[2]	Bulygin, V.S., Nosich, A.I., Gandel, Y.V., “ Nystrom-type method in three-dimensional electromagnetic diffraction by a finite PEC rotationally symmetric surface,” IEEE Transactions on Antennas and Propagation, 60(10), 4710-4718, 2012.
[3]	Nosich, A.A., Gandel, Y.V., Magath, T., Altintas, A., “ Numerical analysis and synthesis of 2D quasi-optical reflectors and beam waveguides based on an integral-equation approach with Nystrom's discretization,” Journal of the Optical Society of America A: Optics and Image Science, and Vision, 24(9), 2831-2836, September 2007.
[4]	Gandel, Y.V., Zaginaylov, G.I., Steshenko, S.A., “Rigorous electrodyanamic analysis of resonator systems of coaxial gyrotrons,” Technical Physics, 49(7), 887-894, July 2004.
[5]	Belotserkovsky, S.M., Lifanov, I.K., Method of Discrete Vortices, CRC Press, New York,1993, 4-464.
[6]	Lifanov, I.K., Singular Integral Equations and Discrete Vortices, VSP, Utrecht, the Netherlands, 1996, 4-475.
[7]	Gandel, Yu.V., “The method of discrete singularities in problems of electrodynamics,” Vopr. Kibern., Moscow, 166-183, 1986. [in. Russian].
[8]	Gandel, Yu.V., Dushkin, V.D., Mathematical models of two-dimensional diffraction problems: Singular integral equations and numerical methods of discrete singularities method, Academy of IT of the MIA of Ukraine, Kharkov, Ukraine, 2012, 4-544. [in. Russian].
[9]	Gandel' Yu.V. “Parametric representations of integral and psevdodifferential operators in diffraction problems,” in 10th Int. Conf. on Math. Methods in Electromagnetic Theory MMET-2004, (Dnepropetrovsk, Ukraine, Sept. 14-17), Dnepropetrovsk, 2004, 57-62.
[10]	Gandel, Yu. V., “Boundary-Value Problems for the Helmholtz Equation and their Discrete Mathematical Models,” Journal of Mathematical Sciences, 171(1), 74-88, 2010.
[11]	Gandel, Y.V., Dushkin, V.D., “The method of parametric representations of integral and pseudo-differential operators in diffraction problems on electrodynamic structures,” in Proceedings of the International Conference Days on Diffraction DD 2012 (28 May-1 June 2012), St. Petersburg, Russian Federation, 2012, 76-81.
[12]	Gandel', Yu.V., Kravchenko, V.F., Pustovoit, V.I. “Scattering of electromagnetic waves by a thin superconducting band,” Doklady Mathematics, 54(3), 959-961, 1996.
[13]	Gandel, Yu.V., Kravchenko, V.F., Morozova, N.N. “Solving the problem of electromagnetic wave diffraction by a superconducting thin stripes grating,” Telecommunications and Radio Engineering (English translation of Elektrosvyaz and Radiotekhnika), 56(2), 15-17, 2001.
[14]	Gandel', Yu.V., Sidel'nikov, G.L., “The method of integral equations in the third boundary value problem of diffraction on a bounded grating over a flat screen,” Differential Equations, 35(9), 1169-1175, Sept. 1999.
[15]	Dushkin, V.D., “Application of the singular integral transform method to the solution of the two-dimensional problem of diffraction of electromagnetic waves from a superconducting layer with rectangular waveguide channels,” Telecommunications and Radio Engineering (English translation of Elektrosvyaz and Radiotekhnika), 56(2), 78-85, 2001.
[16]	Nesvit, K.V. “ Discrete mathematical model of diffraction on pre-cantor set of slits in impedance plane and numerical experiment,” International Journal of Mathematical Models and Methods in Applied Sciences, 7(11), 897-906, 2013.
[17]	Gandel, Yu.V., Polyanskaya, T.S., “Justification of a numerical method for solving systems of singular integral equations in diffraction grating problems,” Differential Equations, 39(9), 1295-1307, September 2003.
[18]	Gandel, Yu.V., Eremenko, S.V., Polyanskaya, T.S. 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. Part II, Kharkov State University, Kharkov, Ukraine, 1992. [in Russian]
[19]	Gandel, Yu.V. Introduction to methods of evaluation of singular and hypersingular integrals,: Publshed by Karazin Kharkiv National University, Kharkov, Ukraine, 2002, 44-62, [in. Russian].
[20]	Kutateladze, S.S., Fundamentals of Functional Analysis, Kluwer Academic Publishers Group, Dordrecht, the Netherlands, 1996, 113-150.
[21]	Natanson, I.P., Constructive Function Theory, Volume 1, Frederic Ungar Puplishing Co., New York, 1964, 128.
[22]	Gabdulkhaev B.G. The optimal approximation of solutions of linear problems, Kazan. Univ. Publishing, Kazan, 1980, 19, [in. Russian].