Integral Equation for the Ahlfors Map on Multiply Connected Regions

Abstract

This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with where is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of we solve for the boundary correspondence function which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

The computation of zeros of Ahlfors map for doubly connected regions

“…Since the Ahlfors map can be written as , where is analytic in and in , Nazar et al [4] have formulated an integral equation for computing as where…”

“…However the integral equation is solved numerically by assuming the zeros of the Ahlfors map are known. In this paper, we extend the approach of [4] to compute the zeros of Ahlfors map for general doubly connected regions.…”

“…Using the integral equation method, most of the conformal mapping of multiply connected regions can be computed efficiently. The integral equation method has been used by many authors to compute the one-to-one conformal mapping from multiply connected regions onto some standard canonical regions [1, 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. In [9], Nasser et al presented a fast multipole method, which is fast and accurate method for numerical conformal mapping of bounded and unbounded multiply connected regions with high connectivity and highly complex geometry.…”

“…In [1], Kerzman and Stein have derived a uniquely solvable boundary integral equation for computing the Szegö kernel of a bounded region and this method has been generalized in [18] In [4], Nazar et al extended the approach of Sangawi [11,15,16] to construct an integral equation for solving where is the boundary correspondence function of Ahlfors map of multiply connected region onto a unit disk. However the integral equation is solved numerically by assuming the zeros of the Ahlfors map are known.…”

“…For computing the Ahlfors map, some integral equations have been given in [4,18,19,20,21,22]. In [1], Kerzman and Stein have derived a uniquely solvable boundary integral equation for computing the Szegö kernel of a bounded region and this method has been generalized in [18] In [4], Nazar et al extended the approach of Sangawi [11,15,16] to construct an integral equation for solving where is the boundary correspondence function of Ahlfors map of multiply connected region onto a unit disk.…”