Journal of Scientific Computing

, Volume 68, Issue 3, pp 1124–1141 | Cite as

Fast Computing of Conformal Mapping and Its Inverse of Bounded Multiply Connected Regions onto Second, Third and Fourth Categories of Koebe’s Canonical Slit Regions

  • Ali W. K. SangawiEmail author
  • Ali H. M. Murid
  • Lee Khiy Wei


This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded multiply connected region onto a disk with spiral slits region \(\varOmega _1\). This extends the methods that have recently been given for mappings onto annulus with spiral slits region \(\varOmega _2\), spiral slits region \(\varOmega _3\), and straight slits region \(\varOmega _4\) but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto all four regions \(\varOmega _1\), \(\varOmega _2\), \(\varOmega _3\), and \(\varOmega _4\) as well as their inverses. The integral equations are solved numerically using combination of Nyström method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is \(O((m + 1)n)\), where \(m+1\) is the multiplicity of the multiply connected region and n is the number of nodes on each boundary component. Previous algorithms require \(O((m+1)^3 n^3)\) operations. The algorithm is tested on several test regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method.


Numerical conformal mapping Boundary integral equations  Multiply connected regions Generalized Neumann kernel GMRES FMM 

Mathematics Subject Classifications

30C30 65E05 



This work was supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia, UTM-CIAM (Vote: R.J130000.7809.4F637), and the Ministry of Higher Education through Department of Mathematics School of Science, University of Sulaimani. These supports are gratefully acknowledged. We wish to thank an anonymous referee for valuable comments and suggestions on the manuscript which improve the presentation of the paper.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Ali W. K. Sangawi
    • 1
    • 2
    Email author
  • Ali H. M. Murid
    • 3
    • 4
  • Lee Khiy Wei
    • 3
    • 4
  1. 1.Department of Mathematics, Faculty of Science and Science Education, School of ScienceUniversity of SulaimaniSulaimaniIraq
  2. 2.Department of Computer, College of Basic EducationCharmo UniversityChamchamal SulaimaniIraq
  3. 3.Department of Mathematics, Faculty of ScienceUniversiti Teknologi MalaysiaJohor BahruMalaysia
  4. 4.UTM Centre for Industrial and Applied MathematicsIbnu Sina Institute for Scientific and Industrial ResearchJohor BahruMalaysia

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