Skip to main content
SpringerLink
Log in

Radial Slit Maps of Bounded Multiply Connected Regions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we present a boundary integral equation method for the numerical conformal mapping of a bounded multiply connected region onto a radial slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Amano, K.: A charge simulation method for the numerical conformal mapping of interior, exterior and doubly connected domains. J. Comput. Appl. Math. 53, 353–370 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

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

    Book  MATH  Google Scholar 

  4. Atkinson, K.E.: A Survey of Numerical Methods for the Solution of Fredholm Integral Equations. SIAM, Philadelphia (1976)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Meiron, D.I., Orszag, S.A., Israeli, M.: Applications of numerical conformal mapping. J. Comput. Phys. 40, 345–360 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Orlando (1984)

    MATH  Google Scholar 

  8. Ellacott, S.W.: On the approximate conformal mapping of multiply connected domains. Numer. Math. 33, 437–446 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gonzalez, M.O.: Classical Complex Analysis. Decker, New York (1992)

    MATH  Google Scholar 

  10. Henrici, P.: Applied and Computational Complex Analysis, vol. 3. Wiley, New York (1974)

    MATH  Google Scholar 

  11. Helsing, J., Ojala, R.: On the evaluation of layer potentials close to their sources. J. Comput. Phys. 227(5), 2899–2921 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, L.-N.: Boundary integral equations approach for numerical conformal mapping of multiply connected regions. PhD thesis, Department of Mathematics, Universiti Teknologi Malaysia (2009)

  13. Kokkinos, C.A., Papamichael, N., Sideridis, A.B.: An orthonormalization method for the approximate conformal mapping of multiply-connected domains. IMA J. Numer. Anal. 10(3), 343–359 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kythe, P.K.: Computational Conformal Mapping. Birkhäuser, New Orleans (1998)

    Book  MATH  Google Scholar 

  15. Murid, A.H.M., Nashed, M.Z., Razali, M.R.M.: Some integral equations related to the Riemann map. In: Papamichael, N., Ruscheweyh, St., Saff, E.B. (eds.) Proceedings of the Third CMFT Conference: Computational Methods and Function Theory 1997, pp. 405–419. World Scientific, Singapore (1999)

    Google Scholar 

  16. Murid, A.H.M., Razali, M.R.M.: An integral equation method for conformal mapping of doubly connected regions. Matematika 15(2), 79–93 (1999)

    Google Scholar 

  17. Murid, A.H.M., Mohamed, N.A.: An integral equation method for conformal mapping of doubly connected regions via the Kerzman-Stein kernel. Int. J. Pure Appl. Math. 38(3), 229–250 (2007)

    MathSciNet  Google Scholar 

  18. Murid, A.H.M., Hu, L.-N.: Numerical experiments on conformal mapping of doubly connected regions onto a disk with a slit. Int. J. Pure Appl. Math. 51(4), 589–608 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Nasser, M.M.S.: Boundary integral equation approach for Riemann problem. PhD thesis, Department of Mathematics, Universiti Teknologi Malaysia (2005)

  20. Nasser, M.M.S.: A boundary integral equation for conformal mapping of bounded multiply connected regions. Comput. Methods Funct. Theory 9(1), 127–143 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

  23. Nasser, M.M.S., Murid, A.H.M., Ismail, M., Alejaily, E.M.A.: Boundary integral equation with the generalized Neumann kernel for Laplace’s equation in multiply connected regions. Appl. Math. Comput. 217, 4710–4727 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nehari, Z.: Conformal Mapping. Dover, New York (1952)

    MATH  Google Scholar 

  25. Okano, D., Ogata, H., Amano, K., Sugihara, M.: Numerical conformal mapping of bounded multiply connected domains by the charge simulation method. J. Comput. Appl. Math. 159, 109–117 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reichel, L.: A fast method for solving certain integral equation of the first kind with application to conformal mapping. J. Comput. Appl. Math. 14(1–2), 125–142 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Saff, E.B., Snider, A.D.: Fundamentals of Complex Analysis. Pearson Education Inc., New Jersey (2003)

    Google Scholar 

  28. Sangawi, A.W.K., Murid, A.H.M., Nasser, M.M.S.: Linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits. Appl. Math. Comput. 218(5), 2055–2068 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Symm, G.T.: Conformal mapping of doubly connected domains. Numer. Math. 13, 448–457 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  30. Trefethen, L.N., Bau, D. III: Numerical Linear Algebra. Philadelphia, SIAM (1997)

    Book  MATH  Google Scholar 

  31. 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 

Download references

Acknowledgements

This work was supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia (FRGS Vot 78479 and GUP Q.J130000.7126.01H75). This support is gratefully acknowledged. We wish to thank an anonymous referee for valuable comments and suggestions on the manuscript which improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Ibnu Sina Institute for Fundamental Science Studies, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

    Ali W. K. Sangawi & Ali H. M. Murid

  2. Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

    Ali W. K. Sangawi & Ali H. M. Murid

  3. Department of Mathematics, College of Science, University of Sulaimani, 46001 UOS, Sulaimani, Kurdistan

    Ali W. K. Sangawi

  4. Department of Mathematics, Faculty of Science, Ibb University, P.O. Box 70270, Ibb, Yemen

    M. M. S. Nasser

  5. Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

    M. M. S. Nasser

Authors
  1. Ali W. K. Sangawi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali H. M. Murid
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. M. S. Nasser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali H. M. Murid.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sangawi, A.W.K., Murid, A.H.M. & Nasser, M.M.S. Radial Slit Maps of Bounded Multiply Connected Regions. J Sci Comput 55, 309–326 (2013). https://doi.org/10.1007/s10915-012-9634-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9634-3

Keywords

Search

Navigation