An integral equation for conformal mapping of multiply connected regions onto a circular region

Abstract. An integral equation is presented for the conformal mapping of multiply connected regions of connectivity m+1 onto a circular region. The circular region is bounded by a unit circle, with centre at the origin, and m number of circles inside the unit circle. The development of theoretical part is based on the boundary integral equation related to a non-homogeneous boundary relationship. An example for verification purpose is given in this paper for the conformal mapping from an annulus onto a doubly connected circular region with centres and radii are assumed to be known

Solving Robin problems in bounded doubly connected regions via an integral equation with the eneralized Neumann Kernel

This paper presents a boundary integral equation method for finding the solution of Robin problems in bounded multiply connected regions. The Robin problems are formulated as a Riemann-Hilbert problems which lead to systems of integral equations and the related differential equations are also constructed that give rise to unique solutions are shown. Numerical results on several test regions are presented to illustrate that the approximate solution when using this method for the Robin problems when the boundaries are sufficiently smooth are accurate

Computing Robin Problem on Unbounded Simply Connected Domain via an Integral Equation with the Generalized Neumann Kernel

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation, and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth

Numerical computation of the conformal map onto lemniscatic domains

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations with the Neumann kernel. This can be done in $O(\ell^2 n \log n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.Comment: Minor revision; simplified Example 6.1, and changed Example 6.2 to a set without symmetr

Fast and accurate computation of the logarithmic capacity of compact sets

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it