Startseite Construction of complex potentials for multiply connected domain
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Construction of complex potentials for multiply connected domain

  • Pyotr N. Ivanshin ORCID logo EMAIL logo
Veröffentlicht/Copyright: 15. Dezember 2020

Abstract

The method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential – an analytic function in an unbounded multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given 𝑛-connected unbounded domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and 𝑛 logarithmic summands. The method is easily computable.

MSC 2010: 30C30; 30C20

References

[1] D. F. Abzalilov and E. A. Shirokova, The approximate conformal mapping onto simply and doubly connected domains, Complex Var. Elliptic Equ. 62 (2017), no. 4, 554–565. 10.1080/17476933.2016.1227978Suche in Google Scholar

[2] N. I. Achieser, Theory of Approximation, Frederick Ungar, New York, 1956. Suche in Google Scholar

[3] U. Böttger, B. Plümper and R. Rupp, Complex potentials, J. Math. Anal. Appl. 234 (1999), no. 1, 55–66. 10.1006/jmaa.1999.6317Suche in Google Scholar

[4] D. Crowdy and J. Marshall, Conformal mappings between canonical multiply connected domains, Comput. Methods Funct. Theory 6 (2006), no. 1, 59–76. 10.1007/BF03321118Suche in Google Scholar

[5] T. K. DeLillo, On some relations among numerical conformal mapping methods, J. Comput. Appl. Math. 19 (1987), no. 3, 363–377. 10.1016/0377-0427(87)90205-6Suche in Google Scholar

[6] T. K. DeLillo, The accuracy of numerical conformal mapping methods: a survey of examples and results, SIAM J. Numer. Anal. 31 (1994), no. 3, 788–812. 10.1137/0731043Suche in Google Scholar

[7] F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. 10.1016/B978-0-08-010067-8.50007-4Suche in Google Scholar

[8] P. N. Ivanshin and E. A. Shirokova, Approximate conformal mappings and elasticity theory, J. Complex Anal. 2016 (2016), Article ID 4367205. 10.1155/2016/4367205Suche in Google Scholar

[9] L. A. Lyusternik and V. I. Sobolev, The Elements of Functional Analysis, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1951. Suche in Google Scholar

[10] A. H. M. Murid and L.-N. Hu, Numerical conformal mapping of bounded multiply connected regions by an integral equation method, Int. J. Contemp. Math. Sci. 4 (2009), no. 21–24, 1121–1147. Suche in Google Scholar

[11] M. M. S. Nasser, A. H. M. Murid and A. W. K. Sangawi, Numerical conformal mapping via a boundary integral equation with the adjoint generalized Neumann kernel, TWMS J. Pure Appl. Math. 5 (2014), no. 1, 96–117. Suche in Google Scholar

[12] A. W. K. Sangawi, A. H. M. Murid and M. M. S. Nasser, Annulus with circular slit map of bounded multiply connected regions via integral equation method, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 4, 945–959. Suche in Google Scholar

[13] R. Schinzinger and P. A. A. Laura, Conformal Mapping, Dover, Mineola, 2003. Suche in Google Scholar

[14] E. A. Shirokova, On the approximate conformal mapping of the unit disk on a simply connected domain, Russian Math. (Iz. VUZ) 58 (2014), no. 3, 47–56. 10.3103/S1066369X14030050Suche in Google Scholar

[15] R. Wegmann, Fast conformal mapping of multiply connected regions, J. Comput. Appl. Math. 130 (2001), no. 1–2, 119–138. 10.1016/S0377-0427(99)00387-8Suche in Google Scholar

[16] R. Wegmann, Methods for numerical conformal mapping, Handbook of Complex Analysis: Geometric Function Theory. Vol. 2, Elsevier Science, Amsterdam (2005), 351–477. 10.1016/S1874-5709(05)80013-7Suche in Google Scholar

[17] R. Wegmann and M. M. S. Nasser, The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions, J. Comput. Appl. Math. 214 (2008), no. 1, 36–57. 10.1016/j.cam.2007.01.021Suche in Google Scholar

[18] A. A. M. Yunus, A. H. M. Murid and M. M. S. Nasser, Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2162, Article ID 20130514. 10.1098/rspa.2013.0514Suche in Google Scholar PubMed PubMed Central

[19] A. A. M. Yunus, A. H. M. Murid and M. M. S. Nasser, Numerical evaluation of conformal mapping and its inverse for unbounded multiply connected regions, Bull. Malays. Math. Sci. Soc. (2) 37 (2014), no. 1, 1–24. Suche in Google Scholar

Received: 2019-05-08
Accepted: 2020-08-03
Published Online: 2020-12-15
Published in Print: 2021-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 25.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jaa-2020-2039/html
Button zum nach oben scrollen