On approximate conformal mapping of a disk and an annulus with radial and circular slits onto multiply connected domains

• Autorzy: Ivanshin Pyotr N. , Shirokova Elena A.

Identyfikatory

DOI

10.17512/jamcm.2020.2.06

• Języki publikacji: EN

Abstrakty

EN

The method of boundary curve reparametrization is generalized to the case of multiply connected domains. We construct the approximate analytical conformal mapping of the unit disk with m circular slits and n-m radial slits and an annulus with (m-1) circular slits and n-m radial slits onto an arbitrary given (n+1) multiply connected finite domain with a smooth boundary. The method is based on extension of the Lichtenstein-Gershgorin equation to a multiply connected domain. The proposed method is reduced to the solution of a linear system with unknown Fourier coefficients. The approximate mapping function has the form of a Cauchy integral. Numerical examples demonstrate that the proposed method is effective in computations.

Słowa kluczowe

EN

conformal mapping; multiply connected domain; Fredholm integral equation

PL

mapowanie konformalne; równanie całkowe Fredholma; domeny połączone

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2020
• Tom: Vol. 19, nr 2
• Strony: 73--84
• Opis fizyczny: Bibliogr. 23 poz., rys.

Twórcy

autor

Ivanshin Pyotr N.

• Kazan Federal University Kazan, Russia

autor

Shirokova Elena A.

• Kazan Federal University Kazan, Russia

Bibliografia

• [1] Schinzinger, R., &Laura, P.A.A. (2003). Conformal Mapping: Methods and Applications. Dover Publications.
• [2] Mityushev, V. (1998). Convergence of the Poincare series for classical Schottky groups. Proc. AMS, 126, 2399-2406.
• [3] Mityushev, V., & Rogosin, S., (1999). Constructive Methods for Linear and Nonlinear Boundary-Value Problems for Analytic Functions. Theory and Applications. Chapman & Hall/CRC.
• [4] Mityushev, V. (2014), Poincare a*-series for classical Schottky groups and its applications, In Milovanovi´c, G.V., Rassias M.Th. (Eds.). Analytic Number Theory, Approximation Theory, and Special Functions. Springer.
• [5] Goluzin, G.M. (1969). Geometric Theory of Functions of a Complex Variable. AMS.
• [6] Murid, A.H.M., & Laey-Nee Hu (2009). Numerical conformal mapping of bounded multiply connected regions by an integral equation method. Int. J. Contemp. Math. Sci., 4, 1121-1147.
• [7] Henrici, P. (1986). Applied and Computational Complex Analysis, Vol. 3. New York: Wiley.
• [8] Wegmann, R. (2001) Fast conformal mapping of multiply connected regions. J. Comput. Appl. Math., 130, 119-138.
• [9] Wegman, R., & Nasser M.M.S. (2008). The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions. J. Comput. Appl. Math., 214, 36-57.
• [10] DeLillo, T.K. (1994). The accuracy of numerical conformal mapping methods: A survey of examples and results. SIAM J. Num. Anal., 31 (3), 788-812.
• [11] DeLillo, T.K. (1987). On some relations among numerical conformal mapping methods. J. Comput. Appl. Math., 19, 363–377.
• [12] Yunus, A.A.M., Murid, A.H.M., & Nasser, M.M.S. (2014). Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and rectilinear slit regions. Proc. of the Royal Society A - Math. Phys. and Eng. Sci., 470(2162), Article No. 20130514.
• [13] Shirokova, E.A., & Ivanshin, P.N. (2016). Approximate conformal mappings and elasticity theory. J. Compl. Analysis, ID 4367205.
• [14] Abzalilov, D.F., & Shirokova, E.A. (2017). The approximate conformal mapping onto simply and doubly connected domains. Complex Variables and Elliptic Equations, 62, 554-565.
• [15] Gakhov, F.D. (1966). Boundary Value Problems. Oxford: Pergamon Press.
• [16] Sangawi, A.W.K., Murid, A.H.M., & Nasser, M.M.S. (2012). Annulus with circular slit map of bounded multiply connected regions via integral equation method. Bull. Malays. Math. Sci. Soc., 35, 945-959.
• [17] Yunus, A.A.M., Murid, A.H.M., & Nasser, M.M.S. (2014). Numerical evaluation of conformal mapping and its inverse for unbounded multiply connected regions. Bull. Malays. Math. Sci. Soc., 1(24), 1-24.
• [18] Wegmann, R. (2005). Methods for Numerical Conformal Mappings. Handbook of Complex Analysis: Geometric Function Theory, 2, Amsterdam, The Netherlands: Elsevier, 351-477.
• [19] Abzalilov, D., & Shirokova, E. (2019). The approximate conformal mapping onto multiply connected domains. Prob. Anal. Issues Anal., 8(1), 3-16.
• [20] von Koppenfels,W., & Stallmann, F. (1959). Praxis der konformen Abbildung. Berlin: Springer-Verlag.
• [21] Shirokova, E.A. (2014). On the approximate conformal mapping of the unit disk on a simply connected domain. Russian Math., 58(3), 47-56.
• [22] El-Shenawy, A., & Shirokova, E.A. (2018). A Cauchy integral method to solve the 2D Dirichlet and Neumann problems for irregular simply-connected domains. Uchenye Zapiski Kazanskogo Universiteta. Ser.Fiz-Mat.N., 160(4), 778-787.
• [23] Nasser, M.M.S. (2011). Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonical slit domains. J. Math. Anal. Appl., 382, 47-56.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).