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Russia Orel
Year
2022
Volume
32
Issue
1
Pages
107-129
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Section Mathematics
Title Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle
Author(-s) Polyanskii I.S.a, Loginov K.O.a
Affiliations The Academy of Federal Security Guard Service of the Russian Federationa
Abstract In the article, an approximate analytical solution of the problem of conformal mapping of internal points of an arbitrary polygon to a unit circle is developed. At the preliminary stage, the conformal mapping problem is formulated as a boundary value problem (Schwartz problem). The latter is reduced to the solution of the Fredholm integral equation of the second kind with a Cauchy-type kernel with respect to an unknown complex density function at the boundary domain, followed by the calculation of the Cauchy integral. The developed approximate analytical solution is based on the Cauchy kernel decomposition in the Legendre polynomial system of the first and second kind. A priori and a posteriori estimates of the convergence and accuracy of the given solution are fulfilled. The exponential convergence of the solution in $L_2\left([0,1]\right)$ and the polynomial one in $C\left([0,1]\right)$ are defined. Calculations on test examples are given for a visual comparison of the effectiveness of the developed solution.
Keywords conformal mapping, arbitrary polygon, Schwartz problem, logarithmic double layer potential, complex density function, Fredholm equation, Legendre polynomials
UDC 517.54
MSC 30C20
DOI 10.35634/vm220108
Received 26 March 2021
Language Russian
Citation Polyanskii I.S., Loginov K.O. Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 107-129.
References
  1. Driscoll T.A., Trefethen L.N. Schwarz-Christoffel mapping, Cambridge: Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511546808
  2. Polyanskii I.S. Baritsentricheskii metod v vychislitel'noi elektrodinamike (Barycentric method in computational electrodynamics), Orel: Russian Federation Security Guard Service Federal Academy, 2017.
  3. Grigor'ev O.A. Numerical-analytical method for conformal mapping of polygons with six right angles, Computational Mathematics and Mathematical Physics, 2013, vol. 53, no. 10, pp. 1447-1456. https://doi.org/10.1134/S0965542513100072
  4. Badreddine M., DeLillo T.K., Sahraei S. A comparison of some numerical conformal mapping methods for simply and multiply connected domains, Discrete and Continuous Dynamical Systems. Ser. B, 2019, vol. 24, no. 1, pp. 55-82. https://doi.org/10.3934/dcdsb.2018100
  5. Polyanskii I.S., Pekhov Yu.S. Barycentric method in solving singular integral equations of the electrodynamic theory of mirror antennas, SPIIRAS Proceedings, 2017, issue 5 (54), pp. 244-262 (in Russian).
  6. Favraud G., Pagneux V. Multimodal method and conformal mapping for the scattering by a rough surface, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015, vol. 471, no. 2175, 20140782. https://doi.org/10.1098/rspa.2014.0782
  7. Radygin V.M., Polyanskii I.S. Modified method of successive conformal mappings of polygonal domains, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2016, no. 1 (39), pp. 25-35 (in Russian). https://doi.org/10.17223/19988621/39/3
  8. Radygin V.M., Polyanskii I.S. Methods of conformal mappings of polyhedra in $\mathbb{R}$$3$ , Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 60-68 (in Russian). https://doi.org/10.20537/vm170106
  9. Nasser M.M.S., Vuorinen M. Conformal invariants in simply connected domains, Computational Methods and Function Theory, 2020, vol. 20, issue 3, pp. 747-775. https://doi.org/10.1007/s40315-020-00351-8
  10. Hao Y., Simoncini V. Matrix equation solving of PDEs in polygonal domains using conformal mappings, Journal of Numerical Mathematics, 2020, vol. 29, no. 3, pp. 221-244. https://doi.org/10.1515/jnma-2020-0035
  11. Trefethen L.N.. Numerical conformal mapping with rational functions, Computational Methods and Function Theory, 2020, vol. 20, issue 3, pp. 369-387. https://doi.org/10.1007/s40315-020-00325-w
  12. Barnett A.H. Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains, SIAM Journal of Scientific Computing, 2014, vol. 36, no. 2, pp. A427-A451. https://doi.org/10.1137/120900253
  13. Shirokova E.A. On the approximate conformal mapping of the unit disk on a simply connected domain, Russian Mathematics, 2014, vol. 58, issue 3, pp. 47-56. https://doi.org/10.3103/S1066369X14030050
  14. Bogatyrev A.B. Conformal mapping of rectangular heptagons, Sbornik: Mathematics, 2012, vol. 203, no. 12, pp. 1715-1735. https://doi.org/10.1070/SM2012v203n12ABEH004284
  15. Wala M., Klöckner A. Conformal mapping via a density correspondence for the double-layer potential, SIAM Journal on Scientific Computing, 2018, vol. 40, no. 6, pp. A3715-A3732. https://doi.org/10.1137/18M1174982
  16. Il'inskii A.S., Polyanskii I.S. An approximate method for determining the harmonic barycentric coordinates for arbitrary polygons, Computational Mathematics and Mathematical Physics, 2019, vol. 59, no. 3, pp. 366-383. https://doi.org/10.1134/S0965542519030096
  17. Gakhov F.D. Kraevye zadachi (Boundary value problems), Moscow: Nauka, 1977.
  18. Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki (Equations of mathematical physics), Moscow: Nauka, 1977.
  19. Muskhelishvili N.I. Singulyarnye integral'nye uravneniya (Singular integral equations), Moscow: Nauka, 1968.
  20. Jung Y., Lim M. Series expansions of the layer potential operators using the Faber polynomials and their applications to the transmission problem, SIAM Journal on Mathematical Analysis, 2021, vol. 53, no. 2, pp. 1630-1669. https://doi.org/10.1137/20M1348698
  21. Kress R. Linear integral equations, New York: Springer, 1999. https://doi.org/10.1007/978-1-4612-0559-3
  22. Gradshtein I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedenii (Tables of integrals, sums, series, and products), Moscow: Fizmatlit, 1963.
  23. Hobson E.W. The theory of spherical and ellipsoidal harmonics, Cambrige: Cambridge University Press, 2012.
  24. Kholshevnikov K.V., Shaidulin V.Sh. On properties of integrals of the Legendre polynomial, Vestnik St. Petersburg University: Mathematics, 2014, vol. 47, no. 1, pp. 28-38. https://doi.org/10.3103/S1063454114010051
  25. Krasnov M.L. Integral'nye uravneniya (Integral equations), Moscow: Nauka, 1975.
  26. Arushanyan I.O. On the numerical solution of boundary integral equations of the second kind in domains with corner points, Computational Mathematics and Mathematical Physics, 1996, vol. 36, no. 6, pp. 773-782. https://zbmath.org/?q=an:1161.65369
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