Computational modeling of hypersingular integral equations for 2D precantor scattering structure
 Abstract
 Keywords
 References

Abstract
This paper presents the investigative study to derive a computational model based on hypersingular integral equations for the preCantor planeparallel diffraction structure. Such structure consists of finite numbers of the thin impedance strips located in the XY plane. A plane transverse magnetic wave is incident from infinity on considered diffraction structure at an angle and need to find the total field resulting from the scattering. The model which is considered in this work is an approximation of real fractal antennas in twodimensional case. Prefractal properties of grating allow producing the newest antennas for modern mobile devices due to their compact size and broadband properties. The purpose of this work is to develop computer model their structure using parametric representation of hypersingular integral operator, Nystrom method with specific quadrature formulas. The numerical results have been obtained and investigated for preCantor structures for calculating physics characteristics. These results have been compared and analyzed in different mathematical models and softwares.

Keywords
Computational Model; Hypersingular Integral Equation; Parametric Representation; PreCantor Grating.

References
[1] A. Sommerfeld, "Mathematische Theorie der Diffraction", Mathematische Annalen.  Springer, Vol.47, No.23, (1896), pp.317374.
[2] J. M. Cowley, Diffraction physics, North Holland}, (1995).
[3] R. W. King, T. T. Wu., The Scattering and Diffraction of Waves, Harvard University Press, (1959).
[4] J. A. Kong, Electromagnetic Wave Theory, EMW Publishing, Cambridge, (2008).
[5] D. Jaggard, On fractal electrodynamics: Recent Advances in Electromagnetic Theory, Springer, (1990).
[6] N. Cohen, "Fractal antenna applications in wireless telecommunications", Electronics Industries Forum of New England, IEEE, (1997), pp.4349.
[7] C. Puente, J. Romeu, R. Pous, A. Cardama, "On the behavior of the Sierpinski multiband fractal antenna", IEEE Transactions on Antennas and Propagation, Vol.46, No.4, (1998), pp.517524.
[8] D. Colton, R. Kress, Integral equation methods in scattering theory, SIAM, (1996).
[9] R. Chapko, R. Kress, L. Monch, "On the Numerical Solution of a hypersingular integral equation for elastic scattering from a planar crack", IMA Journal of Numerical Analysis, Vol. 20, Issue 4, (2000), pp. 601619.
[10] R. Kress, "On the numerical solution of a hypersingular integral equation in scattering theory", Journal of Computational and Applied Mathematics, Vol. 61, Issue 3, (1995), pp. 345360.
[11] L. Farina, P. A. Martin, V. Peron, "Hypersingular integral equations over a disc: convergence of a spectral method and connection with Tranter's method", Journal of Computational and applied mathematics, 269 (2014), pp. 118131.
[12] L. Farina, J. S. Ziebell, "Solutions of hypersingular integral equations over circular domains by a spectral method", Proceedings of the Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Seqeth, (2013), pp. 5266.
[13] M. R. Capobianco, G. Criscuolo, P. Junghanns, "Newton methods for a class of nonlinear hypersingular integral equations", Numerical Algorithms, Vol. 55, Issue 23, (2000), pp. 205221.
[14] Yu. V. Gandel, V. D. Dushkin, Mathematical models 2D diffraction problems: Singular integral equations and numerical methods of discrete singularities, Academy of Interal Troops of the MIA of Ukraine, (2012).
[15] Yu. V. Gandel, "Boundaryvalue problems for the Helmholz equation and their discrete mathematical models", Journal of Mathematical Sciences, Vol.171, No.1, (2010), pp.7488.
[16] L. M. Lytvynenko, S.L. Prosvirnin, Wave Diffraction by Periodic Multilayer Structures, Cambridge Scientific Publishers, (2012).
[17] K. V. Nesvit, Mathematical models of electromagnetic waves diffraction on a planeparallel structures, PhD thesis, (2014).
[18] K. V. Nesvit, "Discrete mathematical model of diffraction on preCantor set of slits in impedance plane and numerical experiment", International Journal of Mathematical Models and Methods in Applied Sciences, Vol.7, No.11, (2013), pp.897906.
[19] K. V. Nesvit, "Investigation of the Scattering of a Plane Electromagnetic Wave on PreFractal Grating and Its Numerical Results", International Journal of Mathematical Models and Methods in Applied Sciences, Vol.8, (2014), pp.489497.
[20] K. V. Nesvit, "Scattering and diffraction of TM modes on a grating consisting of a finite number of prefractal thin impedance strips", Proceedings of the 43rd European Microwave Conference, (2013), pp.11431146.
[21] K. V. Nesvit, "Scattering of TE wave on screened preCantor grating based on hypersingular integral equations", American Journal of Computational and Applied Mathematics, Vol.4, No.1, (2014), pp.916.
[22] K. V. Nesvit, "Diffraction problem of scattering and propagation TM wave on prefractal impedance strips above shielded dielectric layer", International Journal of Applied Mathematical Research, Vol.3, No.1, (2014), pp.714.
[23] K. V. Nesvit, "Mathematical and computational modeling of the diffraction problems by discrete singularities method", AIP Conference Proceedings of the 6th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences, Vol.1629, (2014), pp.102109.
[24] J. D. Jackson, Classical electrodynamics, John Wiley and Sons (WIE), (1998).
[25] I. K. Lifanov, L. N. Poltavskii, G. M Vainikko, Hypersingular Integral Equations and their application, London, New York, Washington: CRC Press, (2004).
[26] Yu. V. Gandel, V. D. Dushkin, "The method of parametric representation of integral and pseudodifferential operator in diffraction problems on electrodynamic structures", Proceedings of the International Conference Days on Diffraction, (2012), pp.7681.
[27] Yu. V. Gandel, "Parametric representations of integral and pseudodifferential operators in diffraction problems", Proceedings of the conference of methematical methods in electromagnetic theory, (2004), pp. 5762.
[28] A. V. Kostenko, "Numerical method for the solution of a hypersingular integral equation of the second kind", Ukrainian Mathematical Journal, Vol. 65, No. 9, (2014), pp. 13731383.
[29] Yu. V. Gandel, Introduction in numerical methods of singular and hypersingular integral equations, Karazin Kharkiv National University, (2002).
[30] Yu. V. Gandel, V. D. Dushkin, "The Approximate method for solving the boundary integral equations of the problem of wave scattering by superconducting lattice", American Journal of Applied Mathematics and Statictics, Vol. 2, No. 6, (2014), pp. 369376.
[31] L. Tsang, J. A. Kong, K. H. Ding, C. Ao., Scattering of Electromagnetic Waves: Numerical Solutions, Wiley Interscience, New York, (2001).

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