Numerical methods for Riemann-Hilbert problems in multiply connected circle domains

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Authors
Balu, Raja
Issue Date
2020-05
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Dissertation
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en_US
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Riemann-Hilbert problems are problems for determining functions analytic in a given domain with speci ed values on the boundary. Since the real and imaginary parts of an analytic function are related by the Cauchy-Riemann equations, both parts cannot be speci ed independently. Riemann-Hilbert problems on multiply connected regions have been studied by several authors in the past. A special kind of Riemann-Hilbert problems on circular regions is necessary for conformal mapping of multiply-connected regions. Wegmann introduced a method of successive conjugation which reduces the general conjugation problem to a sequence of Riemann-Hilbert problems on the circles. Here, we present a new method to solve Riemann-Hilbert problems on the circles. We consider the general conjugation as a Least-Squares problem and use direct and iterative methods to obtain the solution. The resulting linear system has an underlying structure of the form of the identity plus a low rank operator and can be solved e ciently by conjugate gradient-like methods. We present numerical examples and comparisons to the method of Wegmann.

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Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics and Physics
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Wichita State University
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Copyright 2020 by Raja D. Balu All Rights Reserved
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