Numerische Mathematik

, Volume 41, Issue 3, pp 287–307 | Cite as

An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains

  • D. M. Hough
  • N. Papamichael


A numerical method, based on the integral equation formulation of Symm, is described for computing approximations to the mapping functions which accomplish the following conformal maps: (a) the mapping of a domain interior to a closed Jordan curve onto the interior of the unit disc, (b) the mapping of a domain exterior to a closed Jordan curve onto the exterior of the unit disc, (c) the mapping of a doubly-connected domain bounded by two closed Jordan curves onto a circular annulus. The numerical method is based on approximating the unknown source density by cubic splines and “singular” functions, and is particularly suited for the mapping of difficult domains having sharp corners.

Subject classifications

AMS (MOS): 30C30 65R20:CR: 5.18 


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  1. 1.
    Bickley, W.G.: Two-dimensional potential problems for the space outside a rectangle. Proc. London Math. Soc.37, 82–105 (1932)Google Scholar
  2. 2.
    Bowman, F.: Notes on two-dimensional electric field problems. Proc. London Math. Soc.39, 205–215 (1935)Google Scholar
  3. 3.
    Gaier, D.: Integralgleichungen erster Art und konforme Abbildung. Math. Z.147, 113–129 (1976)CrossRefGoogle Scholar
  4. 4.
    Gaier, D.: Das logarithmische Potential und die konforme Abbildung mehrfach zusammenhängender Gebiete. E.B. Christoffel, the influence of his work on Mathematics and the Physical Sciences. Butzer, P.L., Féher, F. (eds.). Basel: Birkhäuser, 1981Google Scholar
  5. 5.
    Hayes, J.K., Kahaner, D.K., Kellner, R.G.: An improved method for numerical conformal mapping. Math. Comput.26, 327–334 (1972)Google Scholar
  6. 6.
    Hough, D.M., Papamichael, N.: The use of splines and singular functions in an integral equation method for conformal mapping. Numer. Math.37, 133–147 (1981)Google Scholar
  7. 7.
    Hough, D.M.: The accurate computation of certain integrals arising in integral equation methods for problems in potential theory. Treatment of integral equations by numerical methods. Baker, C.T.H., Miller, G.F. (eds.). London: Academic Press, 1982 (in press)Google Scholar
  8. 8.
    Jaswon, M.A., Symm, G.T.: Integral Equation Methods in Potential Theory and Elastostatics. London: Academic Press, 1977Google Scholar
  9. 9.
    Lehman, R.S.: Development of the mapping function at an analytic corner. Pacific J. Math.7, 1437–1449 (1957)Google Scholar
  10. 10.
    Papamichael, N., Kokkinos, C.A.: Numerical conformal mapping of exterior domains. Comput. Meths. Appl. Mech. Engrg.31, 189–203 (1982)Google Scholar
  11. 11.
    Papamichael, N., Kokkinos, C.A.: The use of singular functions for the approximate conformal mapping of doubly-connected domains. Technical Report TR/01/82, Dept. of Maths., Brunel University., 1982Google Scholar
  12. 12.
    Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. Princeton University, 1951Google Scholar
  13. 13.
    Symm, G.T.: An integral equation method in conformal mapping. Numer. Math.9, 250–258 (1966)Google Scholar
  14. 14.
    Symm, G.T.: Numerical mapping of exterior domains. Numer. Math.10, 437–445 (1967)Google Scholar
  15. 15.
    Symm, G.T.: Conformal mapping of doubly-connected domains. Numer. Math.13, 448–457 (1969)Google Scholar
  16. 16.
    Wendland, W.L.: On Galerkin collocation methods for integral equations of elliptic boundary value problems. Numerical treatment of integral equations. Albrecht, J., Collatz, L. (eds.) Internat. Ser. Numer. Maths. Basel: Birkhäuser, 1980Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • D. M. Hough
    • 1
  • N. Papamichael
    • 2
  1. 1.Division of MathematicsPolytechnic of the South BankLondonEngland
  2. 2.Department of MathematicsBrunel UniversityUxbridgeEngland

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