^{1, 2, 3}

^{1, 2}

^{4, 5}

^{1}

^{2}

^{3}

^{4}

^{5}

We present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a circular slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized, and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.

In general, the exact conformal mapping functions are unknown except for some special regions. It is well known that every multiply connected regions can be mapped conformally onto the circle with concentric circular slits, the circular ring with concentric circular slits, the circular slit region, the radial slit region, and the parallel slit region as described in Nehari [

The plan of the paper is as follows. Section

Let

Mapping of the bounded multiply connected region

The total parameter domain

Let

It is known that

Lastly, we define integral operators

Suppose that

A complex-valued function

Let

Suppose that

Let

Note that the value of

Combining (

Integral equation methods for computing

Note that, from (

The following theorem from [

The function

By obtaining

This section gives another application of Theorem

Comparison of (

Let

By solving the integral equation (

Since the function

For numerical experiments, we have used some test regions of connectivity two, three, four, and five based on the examples given in [

In this section, we have used three test regions of connectivity one. Only the first test region has known exact mapping function. The results for sup norm error between the exact values of

Error norm (unit circle).

8 | ||

16 | ||

32 |

Consider a region

Mapping a region

Consider the elliptical region bounded by the ellipse

The numerical values of

16 | 3.5383174719052 |

32 | 3.5355590602433 |

64 | 3.5355585660566 |

128 | — |

Mapping for Example

Consider a region

Error norm for Example

8 | |

16 | |

32 | |

64 |

Mapping an original region and its image.

In this section, we have used two test regions of connectivity two whose exact mapping functions are unknown. The first and second test regions are circular frame, and the third test region is bounded by an ellipse and circle. Figures

Error norm for Example

32 | ||

64 | ||

128 | ||

256 |

Error norm for Example

64 | ||

128 | ||

256 |

Error norm for Example

64 | ||

128 | ||

256 |

Mapping a region

Mapping a region

Mapping a region

Consider a pair of circles [

Consider a region

Consider a region

In this section, we have used three test regions of connectivity three. The first test region is bounded by three ellipses, the second test region is bounded by an ellipse and two circles, and the third test region is a circular region. The results for sup norm error between the our numerical values of

Error norm for Example

48 | |||

96 | |||

192 | |||

384 |

Error norm for Example

96 | |||

192 | |||

384 |

The numerical values of

96 | 1.144844712112 | 1.333447560114 | 1.711779222648 |

192 | 1.144844080644 | 1.333446944282 | 1.711778670173 |

384 | — | 1.333446944281 | — |

Let

Mapping a region

Let

Mapping a region

Let

Mapping a region

In this section, we have used four test regions for multiply connected regions whose exact mapping functions are unknown. The results for sup norm error for first and third regions between the our numerical values of

Error norm for Example

64 | ||||

128 | ||||

256 | ||||

512 |

Let

Mapping for Example

Let

The numerical values of

64 | 2.97316998311 | 2.50170500411 | 3.45373711618 | 3.69125205510 |

128 | 2.96757277502 | 2.49923061605 | 3.45041067650 | 3.69904161729 |

256 | 2.96756361086 | 2.49922735100 | 3.45040617845 | 3.69905124306 |

512 | 2.96756361085 | 2.49922735099 | 3.45040617844 | 3.69905124308 |

Error norm for Example

80 | |||||

160 | |||||

320 | |||||

400 | 0 | 0 | 0 |

Mapping a region

Let

Mapping a region

Let

The numerical values of

160 | 0.4081769461 | 0.5470254751 | 0.5470254751 | 0.6850879289 | 0.5258641902 |

320 | 0.4081097591 | 0.5470505181 | 0.5470505181 | 0.6850466360 | 0.5258066821 |

400 | 0.4081097885 | 0.5470505071 | 0.5470505071 | 0.6850466537 | 0.5258067072 |

Mapping a region

In this paper, we have constructed new boundary integral equations for conformal mapping of multiply regions onto a circular slit region. We have also constructed a new method to find the values of modulus of

This work was supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia (FRGS Vote 78479). This support is gratefully acknowledged. The authors would like to thank an anonymous referee for careful reading of the paper and constructive comments and suggestions that substantially improved the presentation of the paper.