Equation Method For Conformal Mapping Research Articles

Numerical Conformal Mapping of Unbounded Multiply Connected Regions onto Circular Slit Regions

This paper presents a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto circular slit regions. Three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the classical and the adjoint generalized Neumann kernels. Several numerical examples are presented.

A Fast Boundary Integral Equation Method for Conformal Mapping of Multiply Connected Regions

This paper presents a fast boundary integral equation method for approximating the conformal mapping from bounded and unbounded multiply connected regions of connectivity $m+1$ onto the canonical region obtained by removing $m$ rectilinear slits from a strip. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the fast multipole method. The presented method requires $O((m+1)n\ln n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component. The numerical results of some test calculations illustrate that our method has the ability to handle regions with complex geometry and very high connectivity. Besides the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's 39 canonical slit regions.

Conformal Mapping of Unbounded Multiply Connected Regions onto Canonical Slit Regions

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels.

An integral equation method for conformal mapping of doubly connected regions involving the neumann kernel

We present an integral equation method for conformal mapping of doubly connected regions onto a unit disc with a circular slit of radius μ $lt$ 1. Our theoretical development is based on the boundary integral equation for conformal mapping of doubly connected region derived by Murid and Razali [15]. In this paper, using the boundary relationship satisfied by the mapping function, a related system of integral equations via Neumann kernel is constructed. For numerical experiment, the integral equation is discretized which leads to a system of linear equations, where μ is assumed known. Numerical implementation on a circular annulus is also presented. Keywords: Conformal mapping; integral equations; doubly connected regions; Neumann kernel.

The use of splines and singular functions in an integral equation method for conformal mapping

We consider the integral equation method of Symm for the conformal mapping of simply-connected domains. For the numerical solution, we examine the use of spline functions of various degrees for the approximation of the source density ?. In particular, we consider ways for overcoming the difficulties associated with corner singularities. For this we modify the spline approximation and in the neighborhood of each corner, where a boundary singularity occurs, we approximate ? by a function which reflects the main singular behaviour of the source density. The singular functions are then blended with the splines, which approximate ? on the remainder of the boundary, so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation. We show, by means of numerical examples, that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy.