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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 970928, 26 pages
http://dx.doi.org/10.1155/2012/970928
Research Article

Circular Slits Map of Bounded Multiply Connected Regions

1Ibnu Sina Institute for Fundamental Science Studies, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia
2Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia
3Department of Mathematics, School of Science, University of Sulaimani, Sulaimani 46001, Iraq
4Department of Mathematics, Faculty of Science, Ibb University, P.O. Box 70270, Ibb, Yemen
5Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Received 14 December 2011; Revised 25 January 2012; Accepted 26 January 2012

Academic Editor: Karl Joachim Wirths

Copyright © 2012 Ali W. K. Sangawi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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 [1, page 334]. Several methods for numerical approximation for the conformal mapping of multiply connected regions have been proposed in [216]. Recently, reformulations of conformal mappings from bounded and unbounded multiply connected regions onto the five canonical slit regions as Riemann-Hilbert problems are discussed in Nasser [12, 13, 17]. An integral equation with the generalized Neumann kernel is then used to solve the RH problem as developed in [18]. The integral equation however involves singular integral which is calculated by Wittich’s method. Murid and Hu [11] formulated an integral equation method based on another form of generalized Neumann kernel for conformal mapping of bounded doubly connected regions onto a disk with circular slit but the kernel of the integral equation involved the unknown circular radii. Discretization of the integral equation yields a system of nonlinear equations which they solved using an optimization method. To overcome this nonlinear problem, Sangawi et al. [19] have developed linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits. In this paper, we describe an integral equation method for computing the conformal mapping function 𝑓 of bounded multiply connected regions onto a circular slit region. This boundary integral equation is constructed from a boundary relationship that relates the mapping function 𝑓 on a multiply connected region with 𝑓, 𝜃(𝑡), and |𝑓|, where 𝜃 is the boundary correspondence function.

The plan of the paper is as follows. Section 2 presents some auxiliary materials. Derivations of two integral equations related to 𝑓 and 𝜃(𝑡) are given in Sections 3 and 4, respectively. Section 5 presents a method to calculate the modulus of 𝑓. In Section 6, we give some examples to illustrate our boundary integral equation method. Finally, Section 7 presents a short conclusion.

2. Notations and Auxiliary Material

Let Ω be a bounded multiply connected region of connectivity 𝑀+1. The boundary Γ consists of 𝑀+1 smooth Jordan curves Γ𝑗, 𝑗=0,1,,𝑀, such that Γ̂𝑗, ̂𝑗=1,,𝑀, lies in the interior of Γ0, where the outer curve Γ0 has counterclockwise orientation and the inner curves Γ̂𝑗, ̂𝑗=1,,𝑀, have clockwise orientation. The positive direction of the contour Γ=𝑀𝑗=0Γ𝑗 is usually that for which Ω is on the left as one traces the boundary (see Figure 1). The curve Γ𝑘 is parametrized by 2𝜋-periodic twice continuously differentiable complex function 𝑧𝑘(𝑡) with nonvanishing first derivative𝑧𝑘(𝑡)=𝑑𝑧𝑘(𝑡)𝑑𝑡0,𝑡𝐽𝑘=[]0,2𝜋,𝑘=0,1,,𝑀.(2.1)

970928.fig.001
Figure 1: Mapping of the bounded multiply connected region Ω of connectivity 𝑀+1 onto a circular slit region.

The total parameter domain 𝐽 is the disjoint union of 𝑀+1 intervals 𝐽0,,𝐽𝑀. We define a parametrization 𝑧 of the whole boundary Γ on 𝐽 by𝑧𝑧(𝑡)=0(𝑡),𝑡𝐽0=[],𝑧0,2𝜋𝑀(𝑡),𝑡𝐽𝑀=[].0,2𝜋(2.2)

Let 𝐻 be the space of all real Hölder continuous 2𝜋-periodic functions 𝜔(𝑡) of the parameter 𝑡 on 𝐽𝑘 for 𝑘=0,1,,𝑀, that is,𝜔𝜔(𝑡)=0(𝑡),𝑡𝐽0,𝜔1(𝑡),𝑡𝐽1,𝜔𝑀(𝑡),𝑡𝐽𝑀.(2.3) Let 𝜃(𝑡) (the boundary corresponding function) be given for 𝑡𝐽 by𝜃𝜃(𝑡)=0(𝑡),𝑡𝐽0,𝜃𝑀(𝑡),𝑡𝐽𝑀.(2.4) Let 𝜇 (a piecewise constant real function) be given for 𝑡𝐽 by𝜇𝜇(𝑡)=0,𝜇1,,𝜇𝑀=𝜇0,𝑡𝐽0,𝜇𝑀,𝑡𝐽𝑀.(2.5) Let 𝐴(𝑡) be a complex continuously differentiable 2𝜋-periodic function for all 𝑡𝐽. The generalized Neumann kernel formed with 𝐴 is defined by1𝑁(𝑡,𝑠)=𝜋Im𝐴(𝑡)𝑧𝐴(𝑠)(𝑠)𝑧(𝑠)𝑧(𝑡).(2.6) The kernel 𝑁 is continuous with1N(𝑡,𝑡)=𝜋12𝑧Im(𝑡)𝑧𝐴(𝑡)Im(𝑡)𝐴(𝑡).(2.7) Define also the kernel 𝑀 by1𝑀(𝑡,𝑠)=𝜋Re𝐴(𝑡)𝑧𝐴(𝑠)(𝑠)𝑧(𝑠)𝑧(𝑡),(2.8) which has a cotangent singularity type (see [18] for more detail). The classical Neumann kernel is the generalized Neumann kernel formed with 𝐴(𝑡)=1, that is,1𝑁(𝑡,𝑠)=𝜋𝑧Im(𝑠)𝑧(𝑠)𝑧(𝑡).(2.9) The adjoint kernel 𝑁(𝑠,𝑡) of the classical Neumann kernel is given by𝑁1(𝑡,𝑠)=𝑁(𝑠,𝑡)=𝜋𝑧Im(𝑡)𝑧(𝑡)𝑧(𝑠).(2.10) The adjoint function to the function 𝐴 is given by𝑧𝐴(𝑡)=(𝑡)𝐴(𝑡)=𝑧(𝑡).(2.11) The generalized Neumann kernel 𝑁(𝑠,𝑡) formed with 𝐴 is given by1𝑁(𝑡,𝑠)=𝜋Im𝐴(𝑡)𝑧𝐴(𝑠)(𝑠)𝑧(𝑠)𝑧(𝑡).(2.12) If 𝐴=1, then𝑁(𝑡,𝑠)=𝑁(𝑡,𝑠).(2.13) We define the Fredholm integral operators 𝐍,𝐍,𝐍 by𝐍𝜐(𝑡)=𝐽𝑁(𝑡,𝑠)𝜐(𝑠)𝑑𝑠,𝑡𝐽,(2.14)𝐍𝜐(𝑡)=𝐽𝐍𝑁(𝑡,𝑠)𝜐(𝑠)𝑑𝑠,𝑡𝐽,(2.15)𝜐(𝑡)=𝐽𝑁(𝑠,𝑡)𝜐(𝑠)𝑑𝑠,𝑡𝐽.(2.16) Note that 𝐍=𝐍, if 𝐴=1.

It is known that 𝜆=1 is an eigenvalue of the kernel 𝑁 with multiplicity 1 and 𝜆=1 is an eigenvalue of the kernel 𝑁 with multiplicity 𝑀 [18]. We define the piecewise constant functions𝜒[𝑗](𝜉)=1,𝜉Γ𝑗,𝑗=0,1,2,,𝑀.0,otherwise.(2.17) Then, we have from [18]𝜒Null(𝐈𝐍)=span{1},Null(𝐈𝐍)=span[1],𝜒[2],,𝜒[𝑀].(2.18)

Lastly, we define integral operators 𝐉 and ̂𝐉 by𝐉𝜐=𝐽12𝜋𝑀𝑗=1𝜒[𝑗](𝑠)𝜒[𝑗]̂(𝑡)𝜐(𝑠)𝑑𝑠,𝐉𝜐=𝐽12𝜋𝑀𝑗=0𝜒[𝑗](𝑠)𝜒[𝑗](𝑡)𝜐(𝑠)𝑑𝑠,(2.19) which are required for uniqueness of solution in a later section.

3. Homogenous and Nonhomogenous Boundary Relationship

3.1. Nonhomogeneous Boundary Relationship for Conformal Mapping

Suppose that 𝑐(𝑧), 𝑄(𝑧), and 𝐻(𝑧) are complex-valued functions defined on Γ such that 𝑐(𝑧)0, 𝐻(𝑧)0, 𝑄(𝑧)0, and 𝐻(𝑧)/(𝑇(𝑧)𝑄(𝑧))satisfies the Hölder condition on Γ. Then, the interior relationship is defined as follows.

A complex-valued function 𝑃(𝑧) is said to satisfy the interior relationship if 𝑃(𝑧) is analytic in Ω and satisfies the nonhomogeneous boundary relationship𝑃(𝑧)=𝑐(𝑧)𝑇(𝑧)𝑄(𝑧)𝐺(𝑧)𝑃(𝑧)+𝐻(𝑧),𝑧Γ,(3.1) where 𝐺(𝑧) analytic in Ω, Hölder continuous on Γ, and 𝐺(𝑧)0 on Γ. The boundary relationship (3.1) also has the following equivalent form:𝐺(𝑧)=𝑐(𝑧)𝑇(𝑧)𝑄(𝑧)𝑃(𝑧)2||||𝑃(𝑧)2+𝐺(𝑧)𝐻(𝑧)𝑃(𝑧),𝑧Γ.(3.2) Let the function 𝐿𝑅(̃𝑧) be defined in the region 𝐶{}Γ by𝐿𝑅1(̃𝑧)=2𝜋iΓ𝑐(̃𝑧)𝐻(𝑤)𝑐(𝑤)(𝑤̃𝑧)𝑄(𝑤)𝑇(𝑤)𝑑𝑤,̃𝑧Ω,(3.3) where Ω is the complement of Ω. The following theorem gives an integral equation for an analytic function satisfying the interior nonhomogeneous boundary relationship (3.1) or (3.2). This theorem generalizes the results of Murid and Razali [9] and can be proved by using the approach used in proving Theorem  3.1 in [20, page 45].

Theorem 3.1. Let 𝑈 and 𝑉 be any complex-valued functions that are defined on Γ. If the function 𝑃(𝑧) satisfies the interior nonhomogeneous boundary relationship (3.1) or (3.2), then 12𝑉(𝑧)+𝑈(𝑧)𝑇(𝑧)𝑄(𝑧)𝑃(𝑧)+PVΓ||||×𝐾(𝑧,𝑤)𝑃(𝑤)𝑑𝑤+𝑐(𝑧)𝑈(𝑧)𝑎𝑗ΩRes𝑤=𝑎𝑗𝑃(𝑤)(𝑤𝑧)𝐺(𝑤)conj=𝑈(𝑧)𝐿𝑅(𝑧),𝑧Γ,(3.4) where 1𝐾(𝑧,𝑤)=2𝜋i𝑐(𝑧)𝑈(𝑧)𝑐(𝑤)𝑤𝑧𝑄(𝑤)𝑉(𝑧)𝑇(𝑤),𝐿𝑤𝑧𝑅(𝑧)=12𝐻(𝑧)1𝑄(𝑧)𝑇(𝑧)+PV2𝜋iΓ𝑐(𝑧)𝐻(𝑤)𝑐(𝑤)(𝑤𝑧)𝑄(𝑤)𝑇(𝑤)𝑑𝑤.(3.5) The symbol “conj” in the superscript denotes complex conjugate, while the minus sign in the superscript denotes limit from the exterior. The sum in (3.4) is over all those zeros 𝑎1,𝑎2,,𝑎𝑀 of 𝐺 that lie inside Ω. If 𝐺 has no zeros in Ω, then the term containing the residue in (3.4) will not appear.

Proof. Suppose that 𝑃(𝑧) and 𝐺(𝑧) are analytic functions in Ω and 𝐺 has a finite number of zeros at 𝑎1,𝑎2,,𝑎𝑀 in Ω. Then, by the calculus of residues, we have 12𝜋𝑖Γ𝑃(𝑤)(𝑤̃𝑧)𝐺(𝑤)𝑑𝑤=𝑎𝑗ΩRes𝑤=𝑎𝑗𝑃(𝑤)(𝑤̃𝑧)𝐺(𝑤),̃𝑧Ω.(3.6) Since 𝑃 and 𝐺 satisfy the Hölder condition on Γ and 𝐺(𝑧)0 on Γ, then 𝑃/𝐺 also satisfies the Hölder condition on Γ. Taking the limit Ω̃z𝑧Γ and applying Sokhotski formula [5], we get 12𝑃(𝑧)1𝐺(𝑧)+PV2𝜋𝑖Γ𝑃(𝑧)(𝑤𝑧)𝐺(𝑤)𝑑𝑤=𝑎𝑗ΩRes𝑤=𝑎𝑗𝑃(𝑤)(𝑤𝑧)𝐺(𝑤),𝑧Γ.(3.7) By taking conjugate to both sides and using (3.1), we get 12𝑃(𝑧)𝑐(𝑧)+1𝑄(𝑧)T(𝑧)2𝐻(𝑧)𝑐(𝑧)1𝑄(𝑧)𝑇(𝑧)PV2𝜋𝑖Γ𝑃(𝑧)𝑐(𝑤)𝑤𝑧𝑄(𝑤)𝑑𝑤1𝑇(𝑤)+PV2𝜋𝑖Γ𝐻(𝑧)𝑑𝑤𝑐(𝑤)𝑤𝑧=𝑄(𝑧)𝑇(𝑧)𝑎𝑗ΩRes𝑤=𝑎𝑗𝑃(𝑤)(𝑤𝑧)𝐺(𝑤)conj,𝑧Γ.(3.8) Multiplying both sides by 𝑐(𝑧) and the fact that 𝑑𝑤=𝑇(𝑤)|𝑑𝑤|, after some arrangement, yield 12𝑃(𝑧)1𝑄(𝑧)𝑇(𝑧)+PV2𝜋𝑖Γ𝑐(𝑧)𝑃(𝑧)𝑐(𝑤)𝑤𝑧||||𝑄(𝑤)𝑑𝑤+𝑐(𝑧)𝑎𝑗ΩRes𝑤=𝑎𝑗𝑃(𝑤)(𝑤𝑧)𝐺(𝑤)conj1=2𝐻(𝑧)1𝑄(𝑧)𝑇(𝑧)+PV2𝜋𝑖Γ𝑐(𝑧)𝐻(𝑧)𝑐(𝑤)(𝑤𝑧)𝑄(𝑧)𝑇(𝑧)𝑑𝑤conj,𝑧Γ.(3.9) Applying Sokhotski formulas again to the expression inside the bracket of the right-hand side yields 12𝑃(𝑧)1𝑄(𝑧)𝑇(𝑧)+PV2𝜋𝑖Γ𝑐(𝑧)𝑃(𝑧)𝑐(𝑤)𝑤𝑧||||𝑄(𝑤)𝑑𝑤+𝑐(𝑧)𝑎𝑗ΩRes𝑤=𝑎𝑗𝑃(𝑤)(𝑤𝑧)𝐺(𝑤)conj=𝐿𝑅(𝑧),𝑧Γ.(3.10) Since 𝑃(𝑧) is analytic in Ω, then by Cauchy integral formula, we have 12𝜋𝑖Γ𝑃(𝑧)𝑤̃𝑧𝑑𝑤=0,𝑧Ω.(3.11) Taking the limit 𝜔̃𝑧𝑧Γ and applying Sokhotiski formulas, we get 121𝑃(𝑧)+PV2𝜋𝑖Γ𝑇(𝑤)𝑃(𝑧)||||𝑤𝑧𝑑𝑤=0,𝑧Γ.(3.12) Multiplying (3.12) by 𝑣(𝑧) and subtracting it from (3.10) multiplied by 𝑢(𝑧) yield (3.4).

3.2. Homogeneous Boundary Relationship for Conformal Mapping

Let 𝑤=𝑓(𝑧) be the analytic function which maps Ω in the 𝑧-plane onto a canonical region of the circular slit region in the 𝑤-plane. Let 0 and 𝑎 be a fixed point in Ω such that 𝑎0. Then, the mapping function is made uniquely determined by assuming that 𝑓(𝑎)=0 and 𝑓(0)= such that the residue of the function 𝑓 at 0 is equal to 1 [1]. Hence, the function 𝑓 can be written in the form1𝑓(𝑧)=𝑧1𝑎𝑒𝑧𝑔(𝑧),(3.13) where 𝑔 is analytic in Ω [12, 13]. Note that the boundary value of 𝑓 can be represented in the form𝑓𝑧𝑝(𝑡)=𝜇𝑝𝑒i𝜃𝑝(𝑡),Γ𝑝𝑧=𝑧𝑝(𝑡),0𝑡𝛽𝑝,𝑝=0,1,,𝑀,(3.14) where 𝜃𝑝 is a boundary correspondence function of Γ𝑝 and 𝜇𝑝 is the radius of the circular slit. The unit tangent to Γ at 𝑧(𝑡) is denoted by 𝑇(𝑧(𝑡))=𝑧(𝑡)/|𝑧(𝑡)|. Thus, it can be shown that||||𝑓(𝑧)=𝑓(𝑧)𝑖||𝜃𝑇(𝑧)𝑝(||𝑡)𝜃𝑝𝑓(𝑡)(𝑧)||𝑓||(𝑧),𝑧Γ.(3.15)

4. Integral Equation Method for Computing 𝐹(𝑍)

Note that the value of 𝜃𝑝(𝑡) may be positive or negative since each circular slit 𝑓(Γ𝑝) is traversed twice. Thus, |𝜃𝑝|/𝜃𝑝=±1. Hence, the boundary relationship (3.15) can be written as||||𝑓(𝑧)=±𝑇(𝑧)𝑓(𝑧)𝑖𝑓(𝑧)||𝑓||(𝑧),𝑧Γ.(4.1) To eliminate the ± sign, we square both sides of the boundary relationship (4.1) to get𝑓(𝑧)2=𝑇(𝑧)2||||𝑓(𝑧)2𝑓(𝑧)2||𝑓||(𝑧)2,𝑧Γ.(4.2) Then, the function 𝐸(𝑧) defined by𝐷(𝑧)=𝑧2𝑓(𝑧)=𝑧2𝑓(𝑧)𝑧𝑔(𝑧)+𝑔(𝑧)𝑒𝑧𝑔(𝑧)(4.3) is analytic in Ω.

Combining (4.3), (4.2), and (3.13), we obtain the following boundary relationship:𝑧𝑒2𝑧(𝑧)𝑎2=𝑧|𝑧|2(𝑎𝑧)2||||𝑓(𝑧)2𝑇(𝑧)2𝐷(𝑧)2||||𝐷(𝑧)2,𝑧Γ.(4.4) Comparison of (4.4) and (3.2) leads to a choice of 𝑃(𝑧)=𝐷(𝑧), 𝑐(𝑧)=𝑧|𝑧|2|𝑓(𝑧)|2/(𝑎𝑧)2, 𝑄(𝑧)=𝑇(𝑧), 𝐺(𝑧)=𝑧𝑒2𝑧(𝑧)/𝑎2, 𝐻(𝑧)=0. Setting 𝑈(𝑧)=𝑇(𝑧)𝑄(𝑧) and 𝑉(𝑧)=1, Theorem 3.1 yields1𝑇(𝑧)𝐷(𝑧)+PV2𝜋iΓ𝑧|𝑧|2||||𝑓(𝑧)2(𝑎𝑤)2𝑇(𝑧)𝑤|𝑤|2||||𝑓(𝑤)2(𝑎𝑧)2𝑤𝑧𝑇(𝑧)||||=𝑤𝑧𝑇(𝑤)𝐷(𝑤)𝑑𝑤𝑧|𝑧|2||||𝑓(𝑧)2𝑎𝑧2𝑇(𝑧)𝑎𝑗ΩRes𝑤=𝑎𝑗𝑎2𝐷(𝑤)(𝑤𝑧)𝑤𝑒2𝑤(𝑤)conj,𝑧Γ.(4.5) Note that 𝑎2𝐷(𝑤)/(𝑤𝑧)𝑤2 has a simple pole at 𝑤=0. To evaluate the residue in (4.5), we use the fact that if 𝐿(𝑧)=𝑑(𝑧)/𝑞(𝑧) where 𝑑(𝑧) and 𝑞(𝑧) are analytic at 𝑧0 and 𝑑(𝑧0)0, 𝑞(𝑧0)=0 and 𝑞(𝑧0)0, which means 𝑧0 is a simple pole of 𝐿(𝑧), thenRes𝑤=𝑧0𝑑𝑧𝐿(𝑤)=0𝑞𝑧0.(4.6) Applying (4.6) to the residue in (4.5) and after several algebraic manipulations, we obtain𝑎𝑗ΩRes𝑤=𝑎𝑗𝑎2𝐷(𝑤)(𝑤𝑧)𝑤𝑒2𝑤𝑔(𝑤)=𝑎2𝑧.(4.7) Thus, integral equation (4.5) becomes 𝐹(𝑍)+Γ𝑁+||||=(𝑧,𝑤)𝐹(𝑤)𝑑𝑤𝑎2𝑧2||||𝑓(𝑧)2𝑎𝑧2𝑇(𝑧),𝑧Γ,(4.8) where𝐹(𝑧)=𝑇(𝑧)𝐷(𝑧),𝐷(𝑧)=𝑧2𝑓𝑁(𝑧),+1(𝑧,𝑤)=2𝜋𝑖𝑇(𝑧)𝑧𝑤𝑧|𝑧|2||||𝑓(𝑧)2(𝑎𝑤)2𝑇(𝑧)𝑤|𝑤|2||||𝑓(𝑤)2(𝑎𝑧)2𝑧𝑤,𝑁+1(𝑡,𝑡)=||𝑧2𝜋(||𝑧𝑡)Im(𝑡)𝑧+1(𝑡)||𝑧𝜋𝑖(||𝑡)𝑧(𝑡)𝑧(𝑡)𝑎𝑧Re(𝑡)1𝑧(𝑡)||𝑧2𝜋𝑖(||𝑧𝑡)(𝑡).𝑧(𝑡)(4.9) By using single valuedness of the mapping function 𝑓 leads to the following condition:12𝜋Γ𝑞𝐹(𝑤)𝑤2||||𝑑𝑤=0,𝑞=0,1,,𝑀.(4.10) By means of Cauchy’s integral formula, we can get the following condition:12𝜋Γ𝐹(𝑤)𝑤||||𝑑𝑤=𝑖.(4.11) Thus, the integral equation (4.8) with the conditions (4.10) and (4.11) should give a unique solution provided the parameters 𝜇𝑝, 𝑝=0,1,,𝑀 that appear in 𝑁+(𝑧,𝑤) are known.

Integral equation methods for computing 𝜇𝑝 and 𝜃𝑝 are discussed in the next two sections.

5. Integral Equation for Computing |𝑓(𝑧)|

Note that, from (3.13) and (3.14), we get the following equation:𝑧||𝑓||||||1(𝑡)𝑔(𝑧(𝑡))=log(𝑧(𝑡))log1𝑧(𝑡)𝑎||||1𝑖arg1𝑧(𝑡)𝑎+𝜃𝑝(𝑡).(5.1) Since 𝑔(𝑧) is analytic in Ω, thus𝐴(𝑡)𝑔(𝑧(𝑡))=𝛾(𝑡)+(𝑡)+𝑖𝜐,(5.2) from (5.1) and (5.2), yields1𝐴(𝑡)=𝑧(𝑡),(5.3)𝛾(𝑡)=log1𝑧(𝑡)𝑎,(5.4)(𝑡)=log𝜇(𝑡)=log𝜇0,log𝜇1,,log𝜇𝑀.(5.5)

The following theorem from [22] gives a method for calculating (𝑡), and hence 𝜇𝑝=|𝑓(𝑧𝑝)|.

Theorem 5.1 (see [22, Theorem 5]). The function is given by =(0,1,,𝑀), where 𝑗=𝛾,𝜙[𝑗]=12𝜋J𝛾(𝑡)𝜙[𝑗](𝑡)𝑑𝑡,(5.6) and where 𝜙[𝑗] is the unique solution of the following integral equation 𝐍𝐈++̂𝐉𝜙[𝑗]=𝜒[𝑗],𝑗=0,1,,𝑀,(5.7) where the kernel 𝑁(𝑠,𝑡) is the adjoint kernel of the kernel 𝑁(𝑠,𝑡) which is formed with 𝐴(𝑡)=𝑧(𝑡).

By obtaining 0,1,,𝑀 from (5.6), in view of (5.5), we obtain𝜇𝑗=𝑒𝑗,𝑗=0,1,,𝑀.(5.8)

6. Integral Equation Method for Computing 𝜃𝑝(𝑡)

This section gives another application of Theorem 3.1 for computing 𝑓/𝑓. Let 𝑓 be the mapping function as described in Section 3.2. Note that (4.2) can be written in the following form:||||𝑓(𝑧)f||||(𝑧)2=𝑇(𝑧)2𝑓(𝑧)𝑓(𝑧)2,𝑧Γ.(6.1) Taking the derivative of both sides of (3.13) together with some elementary calculations yields𝑓(𝑧)+𝑎𝑓(𝑧)𝑧(𝑎𝑧)=𝑧𝑔(𝑧)+𝑔(𝑧).(6.2) Let 𝐸(𝑧)=(𝑓(𝑧)/𝑓(𝑧))+(𝑎/𝑧(𝑎𝑧))=𝑧𝑔(𝑧)+𝑔(𝑧) be analytic in Ω. Then,𝑓(𝑧)𝑎𝑓(𝑧)=𝐸(𝑧)+𝑧(𝑧𝑎),𝑧Γ.(6.3) Equations (6.1) and (6.3) together with some elementary calculations yield𝐸(𝑧)=𝑇(𝑧)2𝐸(𝑧)𝑎𝑇(𝑧)2𝑧𝑧𝑎𝑎𝑧(𝑧𝑎),𝑧Γ.(6.4)

Comparison of (6.4) and (3.1) leads to a choice of 𝑃(𝑧)=𝐸(𝑧), 𝑐(𝑧)=1, 𝑄(𝑧)=𝑇(𝑧), 𝐺(𝑧)=1, 𝐻(𝑧)=(𝑎𝑇(𝑧)2/𝑧(𝑧𝑎))(̄𝑎/̄𝑧(𝑧𝑎)). Setting 𝑈(𝑧)=𝑇(𝑧)𝑄(𝑧) and 𝑉(𝑧)=1, Theorem 3.1 yields1𝐸(𝑧)𝑇(𝑧)+PV2𝜋𝑖Γ𝑇(𝑧)𝑤𝑧𝑇(𝑧)||||𝑤𝑧𝐸(𝑤)𝑇(𝑤)𝑑𝑤=𝑇(𝑧)𝐿𝑅(𝑧),𝑧Γ,(6.5) where𝑇(𝑧)𝐿𝑅1(𝑧)=2𝑎𝑇(𝑧)𝑧(𝑧𝑎)𝑎𝑇(𝑧)𝑧𝑧𝑎1+𝑇(𝑧)PV2𝜋𝑖Γ𝑎1𝑤(𝑤𝑧)(𝑤𝑎)𝑑𝑤𝑇(𝑧)PV2𝜋𝑖Γ𝐴𝑎𝑇(𝑤)2𝑤𝑤𝑎(𝑤𝑧)𝑑𝑤,𝑧Γ.(6.6) Then, it follows from [5, page 91] that1PV2𝜋𝑖Γ𝑎1𝑤(𝑤𝑧)(𝑤𝑎)𝑑𝑤=2𝑎.𝑧(𝑧𝑎)(6.7) From (6.5),(6.6), (6.7), and (6.3), we obtain the integral equation𝑓(𝑧)1𝑓(𝑧)𝑇(𝑧)+PV2𝜋𝑖Γ𝑇(𝑧)𝑧𝑤𝑇(𝑧)𝑧𝑤𝑓(𝑤)||||𝑓(𝑤)𝑇(𝑤)𝑑𝑤=2𝑖Im𝑎𝑇(𝑧)𝑧(𝑧𝑎),𝑧Γ.(6.8) In the above integral equation, let 𝑧=𝑧(𝑡) and 𝑤=𝑧(𝑠). Then, by multiplying both sides of (6.8) by |𝑧(𝑡)| and using the fact that𝑓(𝑧)𝑓(𝑧)𝑧(𝑡)=𝑖𝜃𝑝(𝑡),𝑧Γ,(6.9) the above integral equation can also be written as𝜃𝑝(𝑡)+𝐽𝑁(𝑠,𝑡)𝜃𝑝(𝑠)𝑑𝑠=2Im𝑎𝑧(𝑡)𝑧(𝑡)(𝑧(𝑡)𝑎).(6.10) Since 𝑁(𝑠,𝑡)=𝑁(𝑡,𝑠), the integral equation can be written as an integral equation in operator form𝐈+𝐍𝜃𝑝=𝜓,(6.11) where𝜓=2Im𝑎𝑧(𝑡)𝑧(𝑡)(𝑧(𝑡)𝑎).(6.12) However, 𝜆=1 is an eigenvalue of 𝑁 with multiplicity 𝑀, by [18, Theorem  12]. Therefore, the integral equation (6.11) is not uniquely solvable. To overcome this problem, note that𝐽𝑗𝜃𝑝(𝑡)𝑑𝑡=0,𝑗=1,2,,𝑀,(6.13) which implies𝐉𝜃𝑝=0.(6.14) By adding (6.14) to (6.11), we obtain the integral equation𝐈+𝐍𝜃+𝐉𝑝=𝜓.(6.15) The integral equation (6.15) is uniquely solvable in view of the following theorem which can be proved by using the approach used in proving [22, Theorem  4].

Theorem 6.1. Null𝐈+𝐍=+𝐉{0}.(6.16)

Proof. Let 𝜐Null(𝐈+𝐍+𝐉), that is, 𝜐 is a solution of the integral equation 𝐈+𝐍+𝐉𝜐=0.(6.17) Then, it follows from the definition of the operator 𝐉, (2.18), and the Fredholm alternative theorem that 𝐉=𝐉=𝐉2,𝜒Range(𝐉)=span[1],,𝜒[𝑀]𝜒=Null(𝐈+𝐍),Null(𝐉)=span[1],,𝜒[𝑀]=Null(𝐈+𝐍)=Range𝐈+𝐍.(6.18) Hence, we have 𝐍𝐉=𝐉 and 𝐉𝐍=𝐉𝐍=(𝐍𝐉)=𝐉. By multiplying (6.17) by 𝐉, we obtain 𝐉𝜐=0,𝐈+𝐍𝜐=0.(6.19) Thus, 𝜐Null(𝐉)Null𝐈+𝐍=Range𝐈+𝐍Null𝐈+𝐍.(6.20) Since 𝐴=1, thus the index of the function 𝐴 is given by (see [18] for the definition of the index) 𝜅𝑗=0,𝑗=0,1,,𝑚,𝜅=0.(6.21) The space 𝑆+ defined in [18, Equation (30)] is then given by 𝑆+=span{1}. Then, it follows from [18, Equation (92)] that the dimension of the space 𝑆+ defined in [18, Equation (32)] is given by 𝑆dim(+)=𝑀. Similarly, it follows from [18, Equation (105)] that dimNull𝐈+𝐍𝐍=dimNull𝐈=𝑀.(6.22) Thus, it follows from [18, Lemma 20(b)] that Null(𝐈+𝐍𝑆)=+ and the space 𝑅+𝑆 in [18, Lemma 20(a)] contains only the zero function, that is, 𝑅+𝑆={0}. Thus, it follows from [18, Equation (103)] (applied to the adjoint function 𝐴(𝑡)=𝐴(𝑡)/𝑧(𝑡) instead of 𝐴(𝑡)) and from [18, Equation (100)] that Range𝐈+𝐍Null𝐈+𝐍={0}.(6.23) Hence, it follows from (6.20) that 𝜐=0.

By solving the integral equation (6.15), we get 𝜃𝑝(𝑡). And solving the integral equation (5.7), we get 𝜙[𝑗], 𝑗=0,1,,𝑀, which gives 𝑗 through (5.6) which in turn gives 𝜇𝑗 through (5.8). By solving integral equation (4.8), (4.10), and (4.11) with the known values of 𝜇𝑗, we get 𝐹(𝑧). From the definition of 𝐹(𝑧), we get𝑓(𝑧(𝑡))=𝐹(𝑧(𝑡))𝑧2(𝑡)𝑧.(𝑡)(6.24) Finally, from (3.14) and (6.24), the approximate boundary value of 𝑓(𝑧) is given by||||𝑓(𝑧)=𝑓(𝑧)𝑖||𝜃𝑇(𝑧)𝑝(||𝑡)||𝜃𝑝||𝑓(𝑡)(𝑧)||𝑓||(𝑧),𝑧Γ.(6.25) The approximate interior value of the function 𝑓(𝑧) is calculated by the Cauchy integral formula𝑓(𝑧)=𝑎𝑧1𝑎𝑧2𝜋𝑖Γ𝑎𝑤𝑓(𝑤)1𝑎𝑤𝑤𝑧𝑑𝑤,𝑧Γ.(6.26) For points 𝑧 which are not close to the boundary, the integral in (6.26) is approximated by the trapezoidal rule. However, for the points 𝑧 closed to the boundary Γ, the numerical integration in (6.26) is nearly singular. This difficulty is overcome by using the fact that (1/2𝜋𝑖)Γ(1/(𝑤𝑧))𝑑𝑤=1, and rewrite 𝑓(𝑧) as𝑓(𝑧)=((𝑎𝑧)/𝑎𝑧)(1/2𝜋𝑖)Γ(𝑎𝑤𝑓(𝑤)/(𝑎𝑤))(1/(𝑤𝑧))𝑑𝑤Γ(1/(𝑤𝑧))𝑑𝑤,𝑧Ω.(6.27) This idea has the advantage that the denominator in this formula compensates for the error in the numerator (see [23]). The integrals in (6.27) are approximated by the trapezoidal rule.

7. Numerical Examples

Since the function 𝑧𝑝(𝑡) is 2𝜋-periodic, a reliable procedure for solving the integral equations (6.15), (5.7), and (4.8) with the conditions (4.10) and (4.11) numerically is by using the Nyström’s method with the trapezoidal rule [24]. The trapezoidal rule is the most accurate method for integrating periodic functions numerically [25, page 134–142]. Thus, solving the integral equations numerically reduces to solving linear systems of the form𝐴𝑋=𝐵.(7.1) The above linear system (7.1) is uniquely solvable for sufficiently large number of collocation points on each boundary component, since the integral equations (6.15), (5.7), and (4.8) with the conditions (4.10) and (4.11) are uniquely solvable [26]. The computational details are similar to [6, 1113].

For numerical experiments, we have used some test regions of connectivity two, three, four, and five based on the examples given in [2, 4, 7, 12, 13, 15, 2729]. All the computations were done using MATLAB 7.8.0.347(R2009a)(double precision floating point number). The number of points used in the discretization of each boundary component Γ𝑗 is 𝑛.

7.1. Regions of Connectivity One

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 𝑓, 𝜇1 and approximate values 𝑓𝑛, 𝜇1𝑛 are shown in Table 1.

tab1
Table 1: Error norm (unit circle).

Example 7.1. Consider a region Ω bounded by the unit circle 𝑧Γ(𝑡)=𝑒𝑖𝑡,𝑎=0.2+0.2𝑖,(7.2) Then, the exact mapping function is given by [1, page 340] 𝑔(𝑧)=(𝑎𝑧)𝑎𝑧11𝑎𝑧,𝑟=|𝑎|.(7.3) Figure 2 shows the region and its image based on our method. See Table 1 for results.

fig2
Figure 2: Mapping a region Ω bounded by unit circle onto a circular slit region.

Example 7.2. Consider the elliptical region bounded by the ellipse Γ{𝑧(𝑡)=4cos𝑡+2𝑖sin𝑡},𝑎=0.20.2𝑖.(7.4) Figure 3 shows the region and its image based on our method. See Table 2 for our computed value of 𝜇0.

tab2
Table 2: The numerical values of 𝜇0 for Example 7.2.
fig3
Figure 3: Mapping for Example 7.2.

Example 7.3. Consider a region Ω bounded by 𝑧Γ(𝑡)=(10+3cos3𝑡)𝑒𝑖𝑡,𝑎=0.10.6𝑖.(7.5) Figure 4 shows the region and its image based on our method. See Table 3 for comparison between our computed values of 𝜇0 with those computed values 𝜇0𝑛 of Nasser [12, 13].

tab3
Table 3: Error norm for Example 7.3.
fig4
Figure 4: Mapping an original region and its image.
7.2. Regions of Connectivity Two

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 57 show the region and its image based on our method, and approximate values of 𝜇0 and 𝜇1 are shown in Tables 46.

tab4
Table 4: Error norm for Example 7.4.
tab5
Table 5: Error norm for Example 7.5.
tab6
Table 6: Error norm for Example 7.6.
fig5
Figure 5: Mapping a region Ω bounded by two circles onto a circular slit region.
fig6
Figure 6: Mapping a region Ω bounded by an ellipse and a circle onto a circular slit region.
fig7
Figure 7: Mapping a region Ω bounded by two ellipses onto a circular slit region.

Example 7.4 (circular frame). Consider a pair of circles [28] Γ0𝑧(𝑡)=𝑒𝑖𝑡,Γ1𝑧(𝑡)=0.6+0.2𝑒𝑖𝑡,𝑡0𝑡2𝜋,𝑎=0.25+0.25𝑖,(7.6) such that the region bounded by Γ0 and Γ1 is the region between a unit circle and a circle centered at −0.6 with radius 0.2. Then, Figure 5 shows the region and its image based on our method. See Table 4 for comparison between our computed values of 𝜇0 and 𝜇1 with those computed values 𝜇0𝑛 and 𝜇1𝑛 of Nasser [12, 13].

Example 7.5 (ellipse with one circle). Consider a region Ω bounded by an ellipse and a circle Γ0Γ{𝑧(𝑡)=4cos𝑡+𝑖sin𝑡},1𝑧(𝑡)=1+0.25𝑒𝑖𝑡,𝑡0𝑡2𝜋,𝑎=1.4,(7.7) such that the region bounded by Γ0 and Γ1 is the region between an ellipse and a circle centered at −1 with radius 0.25. Then, Figure 6 shows the region and its image based on our method. See Table 5 for comparison between our computed values of 𝜇0 and 𝜇1 with those computed values 𝜇0𝑛 and 𝜇1𝑛 of Nasser [12, 13].

Example 7.6 (two ellipses). Consider a region Ω bounded by pair of ellipses Γ0Γ{𝑧(𝑡)=4cos𝑡+𝑖sin𝑡},1{𝑧(𝑡)=1+0.7cos𝑡0.3𝑖sin𝑡},𝑡0𝑡2𝜋,𝑎=2.3.(7.8) Figure 7 shows the region and its image based on our method. See Table 6 for comparison between our computed values of 𝜇0 and 𝜇1 with those computed values 𝜇0𝑛 and 𝜇1𝑛 of Nasser [12, 13].

7.3. Regions of Connectivity Three

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 𝜇0, 𝜇1, 𝜇2 and the computed values of 𝜇0𝑛, 𝜇1𝑛, 𝜇2𝑛 obtained from [12, 13] are shown in Tables 79.

tab7
Table 7: Error norm for Example 7.7.
tab8
Table 8: Error norm for Example 7.8.
tab9
Table 9: The numerical values of 𝜇0, 𝜇1, and 𝜇2 for Example 7.9.

Example 7.7 (three ellipses). Let Ω be the region bounded by Γ0Γ{𝑧(𝑡)=10cos𝑡+6𝑖sin𝑡},1Γ{𝑧(𝑡)=42𝑖+3cos𝑡2𝑖sin𝑡},2{𝑧(𝑡)=4+2cos𝑡3𝑖sin𝑡},0𝑡2𝜋,𝑎=7.(7.9) Figure 8 shows the region and its image based on our method. See Table 7 for comparison between our computed values of 𝜇0, 𝜇1, and 𝜇2 with those computed values of Nasser [12].

fig8
Figure 8: Mapping a region Ω bounded by three ellipses onto a circular slit region.

Example 7.8 (ellipse with two circles). Let Ω be the region bounded by [7, 13, 15] Γ0Γ{𝑧(𝑡)=4cos𝑡+𝑖sin𝑡},1Γ{𝑧(𝑡)=1.2+0.3(cos𝑡𝑖sin𝑡)},2{𝑧(𝑡)=1+0.6(cos𝑡𝑖sin𝑡)},0𝑡2𝜋,𝑎=2.50.1𝑖.(7.10) Figure 9 shows the region and its image based on our method. See Table 8 for comparison between our computed values of 𝜇0, 𝜇1, and 𝜇2 with those computed values of Nasser [13].

fig9
Figure 9: Mapping a region Ω bounded by an ellipse and two circles onto a circular slit region.

Example 7.9 (three circles). Let Ω be the region bounded by Γ0𝑧(𝑡)=2𝑒𝑖𝑡,Γ1𝑧(𝑡)=1.2+0.3𝑒𝑖𝑡,Γ2𝑧(𝑡)=1+0.6𝑒𝑖𝑡,0𝑡2𝜋,𝑎=0.51.25𝑖.(7.11) Figure 10 shows the region and its image based on our method. See Table 9 for our computed values of 𝜇0, 𝜇1, and 𝜇2.

fig10
Figure 10: Mapping a region Ω bounded by three circles onto a circular slit region.
7.4. Regions of Connectivity Four and Five

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 𝜇0, 𝜇1, 𝜇2, 𝜇3, 𝜇4 and the computed values of 𝜇0𝑛, 𝜇1𝑛, 𝜇2𝑛, 𝜇3𝑛, 𝜇4𝑛 obtained from [12] are shown in Tables 10 and 12.

tab10
Table 10: Error norm for Example 7.10.

Example 7.10. Let Ω be the region bounded by [12] Γ0𝑧(𝑡)=(10+3cos3𝑡)𝑒𝑖𝑡,Γ1𝑧(𝑡)=3.5+6𝑖+0.5𝑒𝑖𝜋/4𝑒𝑖𝑡+4𝑒𝑖𝑡,Γ2𝑧(𝑡)=5+0.5𝑒𝑖𝜋/4𝑒𝑖𝑡+4𝑒𝑖𝑡,Γ3𝑧(𝑡)=3.56𝑖+0.5𝑒𝑖𝜋/4𝑒𝑖𝑡+4𝑒𝑖𝑡,0𝑡2𝜋,𝑎=8.5+0.1𝑖.(7.12) Figure 11 shows the region and its image based on our method. See Table 10 for comparison between our computed values of 𝜇0, 𝜇1, 𝜇2, and 𝜇3 with those computed values of Nasser [12].

fig11
Figure 11: Mapping for Example 7.10.

Example 7.11 (ellipse with three circles). Let Ω be the region bounded by Γ0Γ{𝑧(𝑡)=2cos𝑡+1.5𝑖sin𝑡},1Γ{𝑧(𝑡)=1+0.25(cos𝑡𝑖sin𝑡)},2Γ{𝑧(𝑡)=1+0.25(cos𝑡𝑖sin𝑡)},3{𝑧(𝑡)=0.75𝑖+0.25(cos𝑡𝑖sin𝑡)},0𝑡2𝜋,𝑎=0.250.25𝑖.(7.13) Figure 12 shows the region and its image based on our method. See Table 11 for our computed values of 𝜇0, 𝜇1, 𝜇2, and 𝜇3.

tab11
Table 11: The numerical values of 𝜇0, 𝜇1, 𝜇2, and 𝜇3 for Example 7.11.
tab12
Table 12: Error norm for Example 7.12.
fig12
Figure 12: Mapping a region Ω bounded by an ellipse and three circles onto a circular slit region.

Example 7.12 (ellipse with four circles). Let Ω be the region bounded by Γ0Γ{𝑧(𝑡)=0.2+8cos𝑡+6𝑖sin𝑡},1Γ{𝑧(𝑡)=3+2𝑖+cos𝑡𝑖sin𝑡},2Γ{𝑧(𝑡)=3+2𝑖+cos𝑡𝑖sin𝑡},3Γ{𝑧(𝑡)=32𝑖+cos𝑡𝑖sin𝑡},4{𝑧(𝑡)=32𝑖+cos𝑡𝑖sin𝑡},0𝑡2𝜋,𝑎=4𝑖.(7.14) Figure 13 shows the region and its image based on our method. See Table 12 for comparison between our computed values of 𝜇0, 𝜇1, 𝜇2, 𝜇3, and 𝜇4 with those computed values of Nasser [12].

fig13
Figure 13: Mapping a region Ω bounded by an ellipse and four circles onto a circular slit region.

Example 7.13 (five ellipses). Let Ω be the region bounded by Γ0Γ{𝑧(𝑡)=1.5𝑖+6cos𝑡+8𝑖sin𝑡},1Γ{𝑧(𝑡)=3+0.5𝑖+1.5cos𝑡𝑖sin𝑡},2Γ{𝑧(𝑡)=3+0.5𝑖+1.5cos𝑡𝑖sin𝑡},3Γ{𝑧(𝑡)=3𝑖+0.7cos𝑡1.7𝑖sin𝑡},4{𝑧(𝑡)=6𝑖+1.7cos𝑡0.7𝑖sin𝑡},0𝑡2𝜋,𝑎=0.4𝑖.(7.15) Figure 14 shows the region and its image based on our method. See Table 13 for our computed values of 𝜇0, 𝜇1, 𝜇2, 𝜇3, and 𝜇4.

tab13
Table 13: The numerical values of 𝜇0, 𝜇1, 𝜇2, 𝜇3, and 𝜇4 for Example 7.13.
fig14
Figure 14: Mapping a region Ω bounded by five ellipses onto a circular slit region.

8. Conclusion

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 𝑓(𝑧). The advantage of our method is that our boundary integral equations are all linear. Several mappings of the test regions of connectivity one, two, three, four, and five were computed numerically using the proposed method. After the boundary values of the mapping function are computed, the interior mapping function is calculated by the means of Cauchy integral formula. The numerical examples presented have illustrated that our boundary integral equation method has high accuracy.

Acknowledgments

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.

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