Boundary integral equation method for conformal mapping of multiply connected regions

Abstract

This work presents two methods for numerical conformal mappings of unbounded multiply connected regions onto several classes of canonical slit regions. The first method is only limited to conformal mapping of unbounded multiply connected regions onto the first category of Koebe's canonical regions. It is based on three boundary integral equations formed with the classical adjoint Neumann kernel, adjoint generalized Neumann kernel and modified Neumann kernel. These integral equations are constructed from a boundary relationship satisfied by an analytic function on the unbounded multiply connected regions. By adding some normalizing conditions, the integral equations are uniquely solvable. The second method is for numerical conformal mapping and its inverse of unbounded multiply connected regions onto the first, second, third and fourth category of Koebe’s canonical regions. It is based on reformulating the conformal mapping problem as a Riemann-Hilbert problem and an adjoint Riemann-Hilbert problem. Two integral equations formed with the adjoint generalized Neumann kernel are constructed. With some normalizing conditions, the integral equations are uniquely solvable. For both methods, discretizing the integral equations with their normalizing conditions lead to systems of linear algebraic equations which are solved by Gaussion elimination method to obtain the boundary values of the mapping functions. The interior values of the mapping functions are then determined by using Cauchy’s integral formula. Cauchy’s integral formula is also used to approximate the interior values of the inverse mapping functions for the second method. Some numerical examples are presented to illustrate the effectiveness of both methods for computing the conformal mappings of unbounded multiply connected regions.