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Numerical Computing of Preimage Domains for Bounded Multiply Connected Slit Domains

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Abstract

In this paper, for a given bounded multiply connected slit domain \(\varOmega \), we present an iterative numerical method for computing a conformally equivalent multiply connected domain G bounded by smooth Jordan curves and the conformal mapping \(w=\varPhi (z)\) from G onto \(\varOmega \). Each iteration of the proposed iterative method requires solving the boundary integral equation with the generalized Neumann kernel. We consider two cases of bounded slit domains, namely the unit disk with radial slit domain and an annulus with radial slit domain. Numerical examples are presented to illustrate that the proposed iterative method converges even for highly connected slit domains.

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References

  1. Amano, K.: A charge simulation method for numerical conformal mapping onto circular and radial slit domains. SIAM J. Sci. Comput. 19, 1169–1187 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aoyama, N., Sakajo, T., Tanaka, H.: A computational theory for spiral point vortices in multiply connected domains with slit boundaries. Jpn. J. Ind. Appl. Math. 30, 485–509 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkinson, K.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  4. Benchama, N., DeLillo, T., Hrycak, T., Wang, L.: A simplified Fornberg-like method for the conformal mapping of multiply connected regions-comparisons and crowding. J. Comput. Appl. Math. 209, 1–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourchtein, L.: Conformal mappings of multiply connected domains onto canonical domains using the Green and Neumann functions. Complex Var. Elliptic Equ. 58(6), 821–836 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crowdy, D.: Analytical solutions for uniform potential flow past multiple cylinders. Eur. J. Mech. B Fluids 25, 459–470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crowdy, D.: Calculating the lift on a finite stack of cylindrical aerofoils. Proc. R. Soc. A 462, 1387–1407 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Crowdy, D.: Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid. J. Eng. Math. 62, 333–344 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crowdy, D.: A new calculus for two-dimensional vortex dynamics. Theor. Comput. Fluid Dyn. 24, 9–24 (2010)

    Article  MATH  Google Scholar 

  10. Crowdy, D.: Conformal slit maps in applied mathematics. ANZIAM J. 53, 171–189 (2012)

    MATH  Google Scholar 

  11. Crowdy, D., Kropf, E.H., Green, C.C., Nasser, M.: The Schottky–Klein prime function: a theoretical and computational tool for applications. IMA J. Appl. Math. 81, 589–628 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Crowdy, D., Marshall, J.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. DeLillo, T.: The accuracy of numerical conformal mapping methods: a survey of examples and results. SIAM J. Numer. Anal. 31, 788–812 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. DeLillo, T., Driscoll, T., Elcrat, A., Pfaltzgraff, J.: Radial and circular slit maps of unbounded multiply connected circle domains. Proc. R. Soc. A 464(2095), 1719–1737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. DeLillo, T., Elcrat, A.: A Fornberg-like conformal mapping method for slender regions. J. Comput. Appl. Math. 46, 49–64 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goluzin, G.: Geometric Theory of Functions of a Complex Variable. Amer. Math. Soc., Rhode Island (1969)

    Book  MATH  Google Scholar 

  17. Greengard, L., Gimbutas, Z.: FMMLIB2D: A MATLAB toolbox for fast multipole method in two dimensions. Version 1.2 (2012). http://www.cims.nyu.edu/cmcl/fmm2dlib/fmm2dlib.html. Accessed 1 Jan 2018

  18. Gutknecht, M.: Numerical experiments on solving theodorsen’s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods. SIAM J. Sci. Stat. Comput. 4(1), 1–30 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Henrici, P.: Applied and Computational Complex Analysis, vol. 3. Wiley, New York (1986)

    MATH  Google Scholar 

  20. Kerzman, N., Trummer, M.: Numerical conformal mapping via the Szegö kernel. J. Comput. Appl. Math. 14, 111–123 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Koebe, P.: Über die konforme Abbildung mehrfach-zusammenhängender Bereiche. Jahresber. Deut. Math. Ver. 19, 339–348 (1910)

    MATH  Google Scholar 

  22. Koebe, P.: Abhandlungen zur theorie der konformen abbildung, iv. abbildung mehrfach zusammenhängender schlichter bereiche auf schlitzbe-reiche. Acta Math. 41, 305–344 (1918)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kress, R.: Linear Integral Equations, 3rd edn. Springer, New York (2014)

    Book  MATH  Google Scholar 

  24. Kropf, E.: http://wwwf.imperial.ac.uk/~ekropf/welding/szego_accuracy.html. Accessed 23 May 2015

  25. Nasser, M.M.S.: A boundary integral equation for conformal mapping of bounded multiply connected regions. Comput. Methods Funct. Theory 9, 127–143 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nasser, M.M.S.: Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel. SIAM J. Sci. Comput. 31(3), 1695–1715 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nasser, M.M.S.: 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 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nasser, M.M.S.: Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions. J. Math. Anal. Appl. 398, 729–743 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nasser, M.M.S.: Fast computation of the circular map. Comput. Methods Funct. Theory 15(2), 187–223 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nasser, M.M.S.: Fast solution of boundary integral equations with the generalized Neumann kernel. Electron. Trans. Numer. Anal. 44, 189–229 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Nasser, M.M.S.: CircularMap: A numerical implementation of the circular map in MATLAB (2017). https://github.com/mmsnasser/CircularMap. Accessed 1 Jan 2018

  32. Nasser, M.M.S., Al-Shihri, F.: A fast boundary integral equation method for conformal mapping of multiply connected regions. SIAM J. Sci. Comput. 35(3), A1736–A1760 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nasser, M.M.S., Green, C.C.: A fast numerical method for ideal fluid flow in domains with multiple stirrers. Nonlinearity 31, 815–837 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Okano, D., Ogata, H., Amano, K.: A method of numerical conformal mapping of curved slit domains by the charge simulation method. J. Comput. Appl. Math. 152, 441–450 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pommerenke, C.: On the logarithmic capacity and conformal mapping. Duke Math. J. 35, 321–325 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Razali, M., Nashed, M., Murid, A.: Numerical conformal mapping via the Bergman kernel. J. Comput. Appl. Math. 82, 335–350 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sangawi, A., Murid, A.H.M., Nasser, M.M.S.: Radial slit maps of bounded multiply connected regions. J. Sci. Comput. 55, 309–326 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Schiffer, M.: Some recent developments in the theory of conformal mapping. In: Appendix to: R. Courant, Dirichlet’s principle, conformal mapping and minimal surfaces. Interscience, New York (1950)

  39. Trummer, M.: An efficient implementation of a conformal mapping method based on the Szegö kernel. SIAM J. Numer. Anal. 23(4), 853–872 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wegmann, R.: Crowding for analytic functions with elongated range. Constr. Approx. 10, 179–186 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wegmann, R.: Methods for numerical conformal mapping. In: Kühnau, R. (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. 2, pp. 351–477. Elsevier, New York (2005)

    Chapter  Google Scholar 

  42. Wegmann, R., Nasser, M.M.S.: The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions. J. Comput. Appl. Math. 214, 36–57 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yunus, A., Murid, A.H.M., Nasser, M.M.S.: Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and rectilinear slit regions. Proc. R. Soc. A. 470(2162), 514 (2014)

    MATH  Google Scholar 

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Acknowledgements

The author is grateful to an anonymous referee for his valuable comments and suggestions which improved the results and the presentation of this paper. Further, the author thanks Prof. Leslie Greengard and Dr. Zydrunas Gimbutas for making the MATLAB toolbox FMMLIB2D [17] publicly available.

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  1. Department of Mathematics, Statistics and Physics, Qatar University, P.O.Box 2713, Doha, Qatar

    Mohamed M. S. Nasser

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  1. Mohamed M. S. Nasser
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Correspondence to Mohamed M. S. Nasser.

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Nasser, M.M.S. Numerical Computing of Preimage Domains for Bounded Multiply Connected Slit Domains. J Sci Comput 78, 582–606 (2019). https://doi.org/10.1007/s10915-018-0784-9

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  • DOI: https://doi.org/10.1007/s10915-018-0784-9

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