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

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Abstract

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.

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Authors and Affiliations

  1. Lobachevskiy Institute of Mathematics and Mechanics, Kazan Federal University, Kremlevskaya St., 35, Kazan, Russia, 420008

    D. F. Abzalilov, P. N. Ivanshin & E. A. Shirokova

Authors
  1. D. F. Abzalilov
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  2. P. N. Ivanshin
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  3. E. A. Shirokova
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Correspondence to D. F. Abzalilov.

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Communicated by Sergey Naboko.

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Abzalilov, D.F., Ivanshin, P.N. & Shirokova, E.A. Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping. Complex Anal. Oper. Theory 13, 1419–1429 (2019). https://doi.org/10.1007/s11785-018-00882-y

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  • DOI: https://doi.org/10.1007/s11785-018-00882-y

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