A numerical method for solving a complete hypersingular integral equation of the second kind and its justification
Abstract
A complete hypersingular integral equation of the second kind was obtained as a boundary integral equation for the diffraction and scattering problem of electromagnetic waves in space separated by the periodically placed non-perfectly conducting strips. The equation includes a singular integral that distinguishes it from the studied second-kind hypersingular equation. Our motivation is the need to have a numerical method for the equation, its applicability borders, and guaranteed convergence. The numerical method has the type of Nyström. The justification of the method envelops a proof of the theorem of existence and uniqueness of the solution and an estimate of the convergence rate of sequence of the approximate solutions to an exact solution.
Keyword : complete hypersingular integral equation, singular integral, integral with the logarithmic kernel, numerical method, Nyström-type, existence, uniqueness, convergence rate, model problem, numerical convergence
How to Cite
Kostenko, O. V. (2023). A numerical method for solving a complete hypersingular integral equation of the second kind and its justification. Mathematical Modelling and Analysis, 28(4), 689–714. https://doi.org/10.3846/mma.2023.14761
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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O.V. Kostenko. A numerical method for solving a system of hypersingular integral equations of the second kind. Cybernetics and Systems Analysis, 52(3):394– 407, 2016. https://doi.org/10.1007/s10559-016-9840-3
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J.L. Tsalamengas. Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods. Journal of Computational Physics, 303:498–513, 2015. https://doi.org/10.1016/j.jcp.2015.09.053
F.O. Yevtushenko, S.V. Dukhopelnykov and A.I. Nosich. H-polarized planewave scattering by a PEC strip grating on top of a dielectric substrate: analytical regularization based on the Riemann-Hilbert problem solution. Journal of Electromagnetic Waves and Applications, 34(4):483–499, 2020. https://doi.org/10.1080/09205071.2020.1722258
T.L. Zinenko, V.O. Byelobrov, M. Marciniak, J. Čtyroký and A.I. Nosich. Grating resonances on periodic arrays of sub-wavelength wires and strips: from discoveries to photonic device applications. In Shulika O. and I. Sukhoivanov (Eds.), Contemporary Optoelectronics. Springer Series in Optical Sciences. Vol. 199, pp. 65–79. Springer, Dordrecht, 2016. https://doi.org/10.1007/978-94-017-7315-7_4
T.L. Zinenko, A. Matsushima and A.I. Nosich. Terahertz range resonances of metasurface formed by double-layer grating of microsize graphene strips inside dielectric slab. Proceedings of the Royal Society A. Mathematical, Physical and Engineering Sciences, 476(2240):1–16, 2020. https://doi.org/10.1098/rspa.2020.0173
V.O. Byelobrov, J. Ctyroky, N.M. Benson, R. Sauleau, A. Altintas and A.I. Nosich. Low-threshold lasing eigenmodes of an infinite periodic chain of quantum wires. Optics Letters, 35(21):3634–3636, 2010. https://doi.org/10.1364/OL.35.003634
M.R. Capobianco, G. Criscuolo and P. Junghanns. A fast algorithm for Prandtl’s integro-differential equation. Journal of Computational and Applied Mathematics, 77(1-2):103–128, 1997. https://doi.org/10.1016/S0377-0427(96)00124-0
M.R. Capobianco, G. Criscuolo and P. Junghanns. On the numerical solution of a hypersingular integral equation with fixed singularities. In T. Ando, R. E. Curto, I. B. Jung and W. Y. Lee(Eds.), Recent Advances in Operator Theory and Applications. Operator Theory: Advances and Applications. Vol. 187, pp. 95–116. Birkha¨user Verlag Basel, Switzerland, 2008. https://doi.org/10.1007/978-3-7643-8893-5_4
S.I. Eminov and V.S. Eminova. Justification of the Galerkin method for hypersingular equations. Computational Mathematics and Mathematical Physics, 56(3):417–425, 2016. https://doi.org/10.1134/S0965542516030039
M. Feischl, T. Fu¨hrer, D. Praetorius and E.P. Stephan. Optimal additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations. Calcolo,54(1):367–399,2017. https://doi.org/10.1007/s10092-016-0190-3
B.G. Gabdulkhaev. Optimal approximations of linear problems. Kazan State University Publishing House, Kazan, 1980. (in Russian)
B.G. Gabdulkhaev. Direct methods for solving singular integral equations of the first kind. Numerical analysis. V. I. Ulyanov Kazan State University Publishing House, Kazan, 1994. (in Russian)
Yu.V. Gandel. Introduction to the methods of calculation of singular and hypersingular integrals. V. N. Karazin Kharkiv National University Publishing House, Kharkiv, 2001. (in Russian)
Yu.V. Gandel and V.D. Dushkin. Mathematical models of two-dimensional diffraction problems: singular integral equations and numerical discrete singularity methods. Publishing House of Academy of Internal Defense of Ministry of Internal Affairs of Ukraine, Kharkiv, 2012. (in Russian)
Yu.V. Gandel, S.V. Eremenko and T.S. Polyanskaya. Mathematical problems of the method of discrete currents. Justification of the numerical method of discrete singularities for solving two-dimensional problems of electromagnetic waves diffraction. Part 2. M. Gorky Kharkiv State University Publishing House, Kharkiv, 1992. (in Russian)
Yu.V. Gandel and A.S. Kononenko. Justification of the numerical solution of a hypersingular integral equation. Differential Equations, 42(9):1256–1262, 2006. https://doi.org/10.1134/S0012266106090114
T. Hartmann and E.P. Stephan. A discrete collocation method for a hypersingular integral equation on curves with corners. In J. Dick, F. Kuo and H. Wo´zniakowski(Eds.), Contemporary Computational Mathematics — A Celebration of the 80th Birthday of Ian Sloan, pp. 545–566. Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-72456-0_25
M.M. Khapaev Jr. Some methods of regularization and numerical solution of integral equations of the first kind. Soviet Mathematics (Izvestiya Vysshikh Uchebnykh Zavedeniy. Matematika), 7:81–85, 1983 [cited January 27, 2021]. Available on Internet: http://www.mathnet.ru/links/9f1f2a276ab6d6f5e00ea82d059da8d6/ivm7042.pdf (in Russian)
V.M. Kadets. Course in functional analysis. V. N. Karazin Kharkiv National University Publishing House, Kharkiv, 2006 [cited January 27, 2021]. Available on Internet: http://page.mi.fu-berlin.de/werner99/kadetsbook/Kadets_Functional_Analysis.pdf (in Russian)
A.V. Kostenko. Numerical method for the solution of a hypersingular integral equation of a second kind. Ukrainian Mathematical Journal, 65(9):1373–1383, 2014. https://doi.org/10.1007/s11253-014-0865-3
O.V. Kostenko. Mathematical models of diffraction by prefractal electrodynamics structures. PhD thesis, A. N. Podgorny Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Kharkiv, 2016. (in Russian)
O.V. Kostenko. A numerical method for solving a system of hypersingular integral equations of the second kind. Cybernetics and Systems Analysis, 52(3):394– 407, 2016. https://doi.org/10.1007/s10559-016-9840-3
O.V. Kostenko. The boundary equations of the diffraction problem of oblique incidence electromagnetic waves on a periodic impedance lattice. In V. O. Marchenko, L. A. Pastur and E. Ya. Khruslov(Eds.), Book of Abstracts of V International Conference “Analysis and Mathematical Physics”, pp. 40–41. B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine Publishing House, Kharkiv, 2017 [cited January 17, 2021]. Available on Internet: http://www.ilt.kharkov.ua/amph2017/AMPH2017_book_of_abstracts.pdf
R. Kress. On the numerical solution of a hypersingular integral equation in scattering theory. Journal of Computational and Applied Mathematics, 61(3):345– 360, 1995. https://doi.org/10.1016/0377-0427(94)00073-7
I.K. Lifanov. Ill-posedness and regularization of the numerical solution of singular integral equations of the first kind. Doklady Akademii Nauk SSSR, 255:1046–1050, 1980 [cited January 25, 2021]. Available on Internet: http://www.mathnet.ru/links/70e74f34da40d0102dd7e1e5e0ba9467/dan44096.pdf (in Russian)
I.K. Lifanov, L.N. Poltavskii and G.M. Vainikko. Hypersingular integral equations and their applications. CRC Press, Boca Raton, 2003. https://doi.org/10.1201/9780203402160
I.P. Natanson. Constructive function theory. Vol. I. Uniform approximation. Frederick Ungar Publishing Company, New York, 1964.
V.V. Panasyuk, M.P. Savruk and Z.T. Nazarchuk. Method of singular integral equations in two-dimensional diffraction problems. Naukova Dumka Publishing House, Kyiv, 1984. (in Ukrainian)
G.N. Pykhteev. Accurate methods for calculating the Cauchy-type integrals. Nauka, Novosibirsk, 1980. (in Russian)
A.V. Setukha. Convergence of a numerical method for solving a hypersingular integral equation on a segment with the use of piecewise linear approximations on a nonuniform grid. Differential Equations, 53(2):234–247, 2017. https://doi.org/10.1134/S0012266117020094
V. Sharma and A. Setia. Numerical solutions and its analysis for a system of hypersingular integral equations. Journal of Computational and Applied Mathematics, 343:520–538, 2018. https://doi.org/10.1016/j.cam.2018.04.052
A. Sidi. Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions. Calcolo, 58(2): Article number 22, 2021. https://doi.org/10.1007/s10092-021-00407-8
J.L. Tsalamengas. Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods. Journal of Computational Physics, 303:498–513, 2015. https://doi.org/10.1016/j.jcp.2015.09.053
F.O. Yevtushenko, S.V. Dukhopelnykov and A.I. Nosich. H-polarized planewave scattering by a PEC strip grating on top of a dielectric substrate: analytical regularization based on the Riemann-Hilbert problem solution. Journal of Electromagnetic Waves and Applications, 34(4):483–499, 2020. https://doi.org/10.1080/09205071.2020.1722258
T.L. Zinenko, V.O. Byelobrov, M. Marciniak, J. Čtyroký and A.I. Nosich. Grating resonances on periodic arrays of sub-wavelength wires and strips: from discoveries to photonic device applications. In Shulika O. and I. Sukhoivanov (Eds.), Contemporary Optoelectronics. Springer Series in Optical Sciences. Vol. 199, pp. 65–79. Springer, Dordrecht, 2016. https://doi.org/10.1007/978-94-017-7315-7_4
T.L. Zinenko, A. Matsushima and A.I. Nosich. Terahertz range resonances of metasurface formed by double-layer grating of microsize graphene strips inside dielectric slab. Proceedings of the Royal Society A. Mathematical, Physical and Engineering Sciences, 476(2240):1–16, 2020. https://doi.org/10.1098/rspa.2020.0173