arXiv:2204.00726v2 [math.CV] 28 Apr 2022 NUMERICAL COMPUTATION OF A PREIMAGE DOMAIN FOR AN INFINITE STRIP WITH RECTILINEAR SLITS EL MOSTAFA KALMOUN, MOHAMED M. S. NASSER, AND MATTI VUORINEN Abstract. Let Ω be the multiply connected domain in the extended complex plane C ob- tained by removing m non-overlapping rectilinear segments from the infinite strip S = {z : | Im z | < π/2}. In this paper, we present an iterative method for numerical computation of a conformally equivalent bounded multiply connected domain G in the interior of the unit disk D and the exterior of m non-overlapping smooth Jordan curves. We demonstrate the utility of the proposed method through two applications. First, we estimate the capacity of condensers of the form (S, E) where E ⊂ S be a union of disjoint segments. Second, we determine the streamlines associated with uniform incompressible, inviscid and irrotational flow past disjoint segments in the strip S. 1. Introduction Let Ω be the multiply connected domain in the extended complex plane C = C∪{∞} obtained by removing m non-overlapping rectilinear segments from the infinite strip S = {z : | Im z| < π/2}. Solving boundary value problems in such a domain with complicated boundaries is not as easy as it is for domains with smooth boundaries. A possible remedy is to find a conformal mapping from Ω onto a multiply connected domain G bordered by smooth Jordan curves. However, up to our knowledge, there is no analytical or numerical method in the literature to compute such a conformal mapping. The above domain Ω is one of the canonical domains for conformal mapping of multiply connected domains [32, p. 128]. An efficient method for numerical computation of the conformal mapping Φ from domains with smooth boundaries G onto the canonical domain Ω is presented in [19]. Still, in this method, the domain G is supposed given while Ω is unknown and should be computed alongside the conformal mapping Φ from G onto Ω. On the contrary, in this paper, we take up the case when Ω is known. Our objective will be then to find an unknown preimage domain G bordered by smooth Jordan curves, and to determine a conformal mapping Φ from G onto Ω. This means that the method presented in [19] is not directly applicable in our context, but this method is still useful to develop an iterative scheme for finding the unknown preimage domain G as well as the conformal mapping Φ. In this way, the inverse mapping Φ −1 would be the desired conformal mapping from the domain Ω onto the domain G. It is known that Laplace equation in the plane is invariant under conformal mappings. Thus, with the help of File: channel-arxiv-v2.tex, printed: 2022-5-2, 3.11 2010 Mathematics Subject Classification. Primary 30C85, 31A15; Secondary 65E05. Key words and phrases. Numerical conformal mappings, condenser capacity, conformal invariance, boundary integral equations. 1