An integral equation method for conformal mapping of doubly connected regions via the Kerzman-Stein and the Neumann Kernels

Abstract

An integral equation method based on the Kerzman-Stein and the Neumann kernels for conformal mapping of doubly connected regions onto an annulus is presented. The theoretical development is based on the boundary integral equations for conformal mapping of doubly connected regions derived by Murid and Razali (1999). However, the integral equations are not in the form of Fredholm integral equation and no numerical experiments are reported. If some information on the zero and singularity of the mapping function is known, then the integral equations can be reduced to the numerically tractable Fredholm integral equations involving the unknown inner radius. For numerical experiments, discretizing the integral equations lead to a system of non-linear equations. The system obtained is solved simultaneously using Newton’s iterative method. Further modification of the integral equations of Murid and Razali (1999) has lead to an efficient and numerically tractable integral equations which involve the unknown inner radius. These integral equations are feasible for all doubly connected regions with smooth boundaries regardless of the information on the zeroes and singularities of the mapping functions. Discretizing the integral equations lead to an over determined system of non-linear equations which is solved using an optimization technique. Numerical implementations on some test regions are also presented.