Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1419–1429 | Cite as

Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping

  • D. F. AbzalilovEmail author
  • P. N. Ivanshin
  • E. A. Shirokova


We present a method for construction of continuous approximate 2D Dirichlet problems solutions in an arbitrary multiply connected domain with a smooth boundary. The method is based on integral equations solution which is reduced to a linear system solution and does not require iterations. Unlike the Fredholm’s solution of the problem ours applies not a logarithsmic potential of a double layer but the properties of Cauchy integral boundary values. We search for the solution of the integral equation in the form of Fourier polynomial with the coefficients being the solution of a linear equation system. The continuous solution of Dirichlet problem is the real part of a Cauchy integral.


Dirichlet problem Conformal mapping Multiply connected domain Fredholm integral equation 



  1. 1.
    Schinzinger, R., Laura, P.A.A.: Conformal Mapping: Methods and Applications. Dover Publications, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Dover Publications, New York (1963)zbMATHGoogle Scholar
  3. 3.
    Tricomi, F.G.: Integral Equations. Dover Publications, New York (1982)Google Scholar
  4. 4.
    Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. A Clarendon Press Publication, Oxford (1986)Google Scholar
  5. 5.
    Shirokova, E.A., El Shenawy, A.: A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains. Numer. Methods Partial Differ. Equ. 34, 2267–2278 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gakhov, F.D.: Boundary Value Problems. Courier Corporation, New York (1990)zbMATHGoogle Scholar
  7. 7.
    Shirokova, E.A., Ivanshin, P.N.: Approximate conformal mappings and elasticity theory. J. Complex Anal. 2016 (2016).
  8. 8.
    Abzalilov, D.F., Shirokova, E.A.: The approximate conformal mapping onto simply and doubly connected domains. Complex Var. Elliptic Equ. 62(4), 554–565 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lyusternik, L.A., Sobolev, V.I.: Elements of Functional Analysis. Nauka, Moscow (1965) (English transl. Wiley, New York, 1974)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Lobachevskiy Institute of Mathematics and MechanicsKazan Federal UniversityKazanRussia

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