Conformal Mapping of Unbounded Multiply Connected Regions onto Canonical Slit Regions

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Volume 2012, Article ID 293765,29pages doi:10.1155/2012/293765

Research Article

Conformal Mapping of Unbounded Multiply Connected Regions onto Canonical Slit Regions

Arif A. M. Yunus,

^{1, 2, 3}

Ali H. M. Murid,

^{1, 2}

and Mohamed M. S. Nasser

^{4, 5}

1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia

2Ibnu Sina Institute for Fundamental Science Studies, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia

3Faculty of Science and Technology, Universiti Sains Islam Malaysia, Negeri Sembilan 71800, Bandar Baru Nilai, Malaysia

4Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

5Department of Mathematics, Faculty of Science, Ibb University, P. O. Box 70270, Ibb, Yemen

Correspondence should be addressed to Ali H. M. Murid,[email protected] Received 22 May 2012; Accepted 18 July 2012

Academic Editor: Matti Vuorinen

Copyrightq2012 Arif A. M. Yunus 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.

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels.

1. Introduction

In this paper, we present a new method for numerical conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. A canonical region in conformal mapping is known as a set of finitely connected regions S such that each finitely connected nondegenerate region is conformally equivalent to a region in S. With regard to conformal mapping of multiply connected regions, there are several types of canonical regions as listed in 1–4. The five types of canonical slit regions are disk with

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concentric circular slits Ud, annulus with concentric circular slits Ua, circular slit regions U_c, radial slit regions U_r, and parallel slit regions U_p. One major setback in conformal mapping is that only for certain regions exact conformal maps are known. One way to deal with this limitation is by numerical computation. Trefethen5has discussed several methods for computing conformal mapping numerically. Amano6and DeLillo et al.7 have successfully map unbounded regions onto circular and radial slit regions. Boundary integral equation related to a boundary relationship satisfied by a function which is analytic in a simply or doubly connected region bounded by closed smooth Jordan curves has been given by Murid8and Murid and Razali9. Special realizations of this integral equation are the integral equations related to the Szeg ¨o kernel, Bergmann kernel, Riemann map, and Ahlfors map. The kernels arise in these integral equations are the Neumann kernel and the Kerzman-Stein kernel. Murid and Hu 10 managed to construct a boundary integral equation for numerical conformal mapping of a bounded multiply connected region onto a unit disk with slits. However, the integral equation involves unknown radii which lead to a system of nonlinear algebraic equations upon discretization of the integral equation.

Nasser11–13produced another technique for numerical conformal mapping of bounded and unbounded multiply connected regions by expressing the mapping function in terms of the solution of a uniquely solvable Riemann-Hilbert problem. This uniquely solvable Riemann Hilbert problem can be solved by means of boundary integral equation with the generalized Neumann kernel. Recently, Sangawi et al. 14–17 have constructed new linear boundary integral equations for conformal mapping of bounded multiply region onto canonical slit regions, which improves the work of Murid and Hu 10 where in10, the system of algebraic equations are nonlinear. In this paper, we extend the work of14–17 for numerical conformal mapping of unbounded multiply connected regions onto all five types of canonical slit regions. These boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region.

The plan of this paper is as follows: Section 2 presents some auxiliary material.

Section 3 presents a boundary integral equation related to a boundary relationship. In Sections4–8, we present the derivations for numerical conformal mapping for all five types of canonical regions. InSection 9, we give some examples to illustrate the eﬀectiveness of our method. Finally,Section 10presents a short conclusion.

2. Auxiliary Material

Let Ω⁻ be an unbounded multiply connected region of connectivity m. The boundary Γ consists ofmsmooth Jordan curvesΓj,j1,2, . . . , mand will be denoted byΓ Γ1∪Γ2∪ · · · ∪ Γm. The boundariesΓj are assumed in clockwise orientationseeFigure 1 . The curveΓj is parameterized by 2π-periodic twice continuously diﬀerentiable complex functionη_jt with nonvanishing first derivative, that is,

η_jt dη_jt

dt /0, t∈J_j 0,2π, k1, . . . , m. 2.1

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ΓM

Γ2

Γ1

Ω⁻

Figure 1: An unbounded multiply connected regionΩ⁻with connectivitym.

The total parameter domain J is the disjoint union of m intervals J1, . . . , Jm. We define a parameterizationηof the whole boundaryΓonJby

ηt

⎧⎪

⎪⎨

⎪⎪

⎩

η₁t , t∈J₁ 0,2π, ...

ηmt , t∈Jm 0,2π.

2.2

Let Φz be the conformal mapping function that maps Ω⁻ onto U⁻, where U⁻ represents any canonical region mentioned above,z₁ is a prescribed point located insideΓ1, z_mis a prescribed point insideΓmandβis a prescribed point located inΩ⁻. In this paper, we determine the mapping functionΦz by computing the derivatives of the mapping function Φηt and two real functions on J, that is, the unknown function ϕt and a piecewise constant real function Rt . LetH be the space of all real H ¨older continuous 2π-periodic functions andSbe the subspace ofHwhich contains the piecewise real constant functions Rt . The piecewise real constant functionRt can be written as

Rt

⎧⎪

⎪⎨

⎪⎪

⎩

R1, t∈J1 0,2π, ...

Rm, t∈Jm 0,2π,

2.3

briefly written asRt R1, . . . , R_m . LetAt be a complex continuously diﬀerentiable 2π- periodic function for allt∈J. We define two real kernels formed withAas18

Ns, t 1 πIm

As At

ηt ηt −ηs

,

Ms, t 1 π Re

As At

ηt ηt −ηs

.

2.4

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The kernelNs, t is known as the generalized Neumann kernel formed with complex-valued functionsAandη. The kernelNs, t is continuous with

Nt, t 1 πIm

ηt

ηt −At At

. 2.5

The kernelMs, t has a cotangent singularity

Ms, t − 1

2πcots−t

2 M1s, t , 2.6 where the kernelM1s, t is continuous with

M₁t, t 1 π Re

1 2π

ηt

ηt −At At

. 2.7

The adjoint functionAofAis defined by

A ηt

At . 2.8

The generalized Neumann kernelNs, t and the real kernelMformed withA are defined by

Ns, t 1 π Im

As At

ηt ηt −ηs

,

Ms, t 1 πRe

As At

ηt ηt −ηs

.

2.9

Then,

Ns, t −N^∗s, t , Ms, t −M^∗s, t , 2.10 whereN^∗s, t Nt, s is the adjoint kernel of the generalized Neumann kernelNs, t . We define the Fredholm integral operators N^∗by

N^∗υt

J

N^∗t, s υs ds, t∈J. 2.11

Integral operatorsM^∗,N, andMare defined in a similar way. Throughout this paper, we will assume the functionsAandAare given by

At 1, At ηt . 2.12

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It is known thatλ1 is not an eigenvalue of the kernelNandλ−1 is an eigenvalue of the kernelNwith multiplicitym18. The eigenfunctions ofNcorresponding to the eigenvalue λ−1 are{χ¹, χ², . . . , χ^m}, where

χ^jξ

1, ξ∈Γj,

0, otherwise. j1,2, . . . , m. 2.13

We also define an integral operator J bysee14

Jμs :

J

1 2π

m j0

χ^js χ^jt μt dt. 2.14

The following theorem gives us a method for calculating the piecewise constant real function ht in connection with conformal mapping later. This theorem can be proved by using the approach as in19, Theorem 5.

Theorem 2.1. Let i√

−1,γ,μ∈Handh∈Ssuch that

Af γhiμ 2.15

are the boundary values of a functionfz analytic inΩ⁻. Then the functionh h1, h2, . . . , hm has each element given by

hj γ, ρ^j

1 2π

Γγt ρ^tdt, 2.16 whereρ^tis the unique solution of the integral equation

IN^∗J ρ^j−χ^j, j 1,2, . . . , m. 2.17

3. The Homogeneous Boundary Relationship

Suppose we are given a functionDz which is analytic inΩ⁻, continuous onΩ⁻∪Γ, H ¨older continuous onΓandD∞ is finite. The boundaryΓjis assumed to be a smooth Jordan curve.

The unit tangent toΓat the pointηt ∈Γwill be denoted byTηt ηt /|ηt |. Suppose further thatDηt satisfies the exterior homogeneous boundary relationship

D ηt

ct T ηt ²

D ηt P

ηt , 3.1

wherect andPare complex-valued functions with the following properties:

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1 Pz is analytic inΩ⁻and does not have zeroes onΩ⁻∪Γ, 2 P∞ /0,D∞ is finite,

3 ct /0,Pηt /0.

Note that the boundary relationship3.1 also has the following equivalent form:

P ηt

ct T

ηt 2 D ηt ₂ D

ηt ². 3.2

Under these assumptions, an integral equation forDηt can be constructed by means of the following theorem.

Theorem 3.1. If the functionDηt satisfies the exterior homogeneous boundary relationship3.1 , then

φt

J

Ks, t φs dsνt , 3.3

where

φt D ηt

ηt ,

Ks, t 1 2πi

⎡

⎢⎣ ηt

ηt −ηs − ct cs

ηt ηt −ηs

⎤

⎥⎦,

νt D∞ ηt ct ηt D∞

P∞ .

3.4

Proof. Consider the integralI1ηt ,

I₁ ηt

1 2πi

J

D ηs

ηs −ηt ds. 3.5

Since the boundary is in clockwise orientation andDis analytic inΩ⁻, then by20, p. 2we have

I₁ ηt

D ηt

2 −D∞ . 3.6

Now, let the integralI2ηt be defined as

I2

ηt 1

2πi

J

ct T ηt ²

D ηs cs

ηs −ηt T

ηs |ds|. 3.7

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Using the boundary relationship3.1 and the fact thatTηt |dt|dtandPηt does not contain zeroes, then by20, p. 2we obtain

I₂ ηt

−ct T

ηt ²D ηt P

ηt ct T

ηt ²D∞

P∞ . 3.8

Next, by takingI₂ηt −I₁ηt with further arrangement yields

D ηt

1 2πi

J

⎡

⎣ 1

ηt −ηs − ct cs

T ηt ² ηt −ηs

⎤

⎦D ηs

|ds|

D∞ ct T

η²D∞

P∞ .

3.9

Then multiplying3.9 withTηt and|ηt |, subsequently yields3.3 . Theorem 3.2. The kernelKs, t is continuous with

Kt, t 1

2πImηt ηt − 1

2πi ct

ct . 3.10

Proof. Let the kernelKs, t be written as

Ks, t K1s, t K2s, t , 3.11

where

K₁s, t 1 2πi

ηt

ηt −ηs − ηt ηt −ηs

,

K₂s, t 1 2πi

−ct cs

ηt

ηt −ηs ηt ηt −ηs

1

2πi ηt cs

cs −ct ηt −ηs

.

3.12

Notice thatK₁s, t is the classical Neumann kernel withK₁t, t 1/2π Imηt /ηt . Now forK₂s, t , as we take the limits → twe have,

K2t, t 1 2πilim

s→t

ηt cs lim

s→t

cs −ct ηt −ηs

− 1 2πi

ct ct .

3.13

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Hence, by combiningK1t, t andK2t, t , we obtain Kt, t 1

2π

Imηt ηt −1

i ct

ct

. 3.14

Note that whenct 1, the kernelKs, t reduces to the classical Neumann kernel.

We define the Fredholm integral operator K by

Kυt 1 2πi

J

Ks, t υs ds, t∈J. 3.15

Hence,3.3 becomes

IK φt νt . 3.16

The solvability of the integral equation 3.16 depends on the possibility of λ −1 being an eigenvalue of the kernel Ks, t . For the numerical examples considered in this paper, λ −1 is always an eigenvalue of the kernelKs, t . Although there is no theoretical proof yet, numerical evidence suggests thatλ −1 is an eigenvalue ofKs, t . If the multiplicity of the eigenvalueλ −1 ism, then one need to add m conditions to the integral equation to ensure the integral equation is uniquely solvable.

4. Exterior Unit Disk with Circular Slits

The canonical regionUd is the exterior unit disk along withm−1 arcs of circles. We assume thatw Φz maps the curveΓ1onto the unit circle|w|1, the curveΓj, wherej 2,3, . . . , m, onto circular slit on the circle|w|R_j, whereR₂, R₃, . . . , R_mare unknown real constants. The circular slits are traversed twice. The boundary values of the mapping functionΦare given by

Φ ηt

Rt e^iθt, 4.1 whereθt represents the boundary correspondence function andRt 1, R2, . . . , Rm . By diﬀerentiating4.1 with respect to tand dividing the result obtained by its modulus, we have

Φ ηt

ηt Φ

ηt

ηt i sign θt

e^iθt. 4.2

Using the fact that unit tangentTηt ηt /|ηt | and e^iθt Φηt /Rt , it can be shown that

Φ ηt

sign

θt Rt i T

ηt Φ ηt Φ

ηt . 4.3

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Boundary relationship4.3 is useful for computing the boundary values ofΦz provided θt , Rt , andΦηt are all known. By taking logarithmic derivative on4.1 , we obtain

ηt Φ ηt Φ

ηt iθt . 4.4

The mapping functionΦz can be uniquely determined by assuming Φ∞ ∞, c Φ∞ lim

z→ ∞

Φz

z >0. 4.5

Thus, the mapping function can be expressed as12

Φz cz−z1 e^Fz, 4.6

whereFz is an analytic function andF∞ 0. By taking logarithm on both sides of4.6 , we obtain

F ηt

lnRt

c iθ−log

ηt −z1

. 4.7

Hence4.7 satisfies boundary values2.15 inTheorem 2.1withAt 1, ht

ln1

c,lnR2

c , . . . ,lnRm

c

, γt −lnηt −z1. 4.8

Hence, the values ofR_jcan be calculated by

R_je^h^j^−h¹ forj 1,2, . . . , m. 4.9 To findθt , we began by diﬀerentiating4.7 and comparing with4.4 which yields

iθt F ηt

ηt − ηt

ηt −z₁. 4.10

In view ofA ηt and lettingfz Fz −1/z−z1 , wherefz is analytic function in Ω⁻,4.10 becomes

Af ηt

iθt . 4.11

By18, Theorem 2c , we obtainI−N θ 0 which implies

IN^∗ θ0. 4.12

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However, this integral equation is not uniquely solvable according to18, Theorem 12. To overcome this, since the image of the curveΓ1 is clockwise oriented and the images of the curvesΓj, j 2,3, . . . , mare slits so we haveθ₁2π −θ₁0 −2π andθ_j2π −θ_j0 0, which implies

Jθt ht −1, 0, . . . ,0 . 4.13

By adding this condition to4.12 , the unknown functionθt is the unique solution of the integral equation

IN^∗J θt ht . 4.14 Next, the presence of signθt in4.3 can be eliminated by squaring both sides of4.3 , that is,

Φ ηt ₂

−Rt ²T

ηt ₂ Φ ηt ₂ Φ

ηt ². 4.15 Upon comparing4.15 with3.2 , this leads to a choice ofPηt Φηt ²,P∞ ∞, Dηt Φηt ,D∞ c,ct −Rt ². Hence,

φt Φ ηt

ηt 4.16

satisfies the integral equation3.16 with

νt Φ∞ ηt . 4.17

Numerical evidence shows thatλ −1 is an eigenvalue ofKs, t of multiplicitym, which means one needs to addmconditions. SinceΦz is assumed to be single-valued, it is also required that the unknown mapping functionΦz satisfies4

Jj

φt dt

Γj

Φ η

dη0, j1,2, . . . , m, 4.18

that is,

Jφ0. 4.19

Then,φt is the unique solution of the following integral equation

IKJ φt νt . 4.20

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By obtainingφt , the derivatives of the mapping function,Φt can be found using Φt φt

ηt . 4.21

By obtaining the values of Rt , θt and Φηt , the boundary value of Φηt can be calculated by using4.3 .

5. Annulus with Circular Slits

The canonical regionU_a consists of an annulus centered at the origin together with m−2 circular arcs. We assume thatΦz maps the curveΓ1onto the unit circle|Φ| 1, the curve Γmonto the circle|Φ|RmandΓjonto circular slit|Φ|Rj, wherej 2,3, . . . , m−1. The slit are traversed twice. The boundary values of the mapping functionΦare given by

Φ ηt

Rt e^iθt, 5.1 whereθt represents the boundary correspondence function andRt 1, R2, . . . , R_m is a piecewise real constant function. By using the same reasoning as inSection 4, we get

Φ ηjt

sign

θt Rjt i T

ηjt Φ ηjt Φ

η_jt , 5.2

ηt Φ ηt Φ

ηt iθt . 5.3

The mapping functionΦz can be uniquely determined by assumingc Φ∞ > 0. Thus, the mapping function can be expressed as12

Φz cz−zm

z−z₁e^Fz. 5.4

By taking logarithm on both sides of5.4 , we obtain F

ηt

lnRt

c iθ−logηt −zm

ηt −z1

. 5.5

Hence,5.5 satisfies boundary values2.15 inTheorem 2.1withAt 1, ht

ln1

c,lnR2

c , . . . ,lnRm

c

, γt −ln

ηt −z_m ηt −z₁

. 5.6 By obtaininghj, the values ofRjcan be computed by

Rje^h^j^−h¹, j 2,3, . . . , m. 5.7

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By diﬀerentiating5.5 and comparing with5.3 yields

iθt F ηt

ηt ηt

ηt −z_m − ηt

ηt −z₁. 5.8

In view ofAηt andfηt Fηt 1/ηt −zm −1/ηt −z1 , wherefz is analytic inΩ⁻,5.8 is equivalent to

Af ηt

iθt . 5.9

By18, Theorem 2c , we obtain

IN^∗ θt 0. 5.10

Note that the image of the curve Γ1 is counterclockwise oriented,Γm is clockwise oriented and the images of the curvesΓj, j 2,3, . . . , m−1 are slits so we haveθ₁2π −θ₁0 2π, θm2π −θm0 −2πandθj2π −θj0 0, which implies

Jθt ht 1, 0, . . . ,−1 . 5.11

Hence, the unknown functionθt is the unique solution of the integral equation

IN^∗J θt ht . 5.12

Next, the presence of signθt in 5.2 can be eliminated by squaring both sides of the equation, that is,

Φ ηt ₂

−Rt ²T

ηt ₂ Φ ηt ₂ Φ

ηt ². 5.13 Comparing5.13 with3.2 leads to a choice ofPηt Φηt ²,P∞ c²,Dηt Φηt ,D∞ 0,ct −Rt ². Hence,

φt Φ ηt

ηt 5.14

νt 0, . . . ,0 . 5.15

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Numerical evidence shows thatλ−1 is an eigenvalue ofKs, t of multiplicitym1, which implies one needs to addm1 conditions. SinceΦz is assumed to be single-valued, hence it is also required that the unknown mapping functionΦz satisfies4

Jj

φs ds

Γj

Φ η

dη0, j1,2, . . . , m, 5.16

that is,

Jφ0. 5.17

Since we assume the mapping functionΦz can be uniquely determined byc Φ∞ >0, hence by20

Φ∞ c 1 2π

J

ηt θt

η−z₁φt dt. 5.18

If we define the Fredholm operator G as

Gμs 1 2π

J

ηt θt

η−z1

μt dt, 5.19

thenφt is the unique solution of the following integral equation:

IKJG φt c. 5.20

By obtainingφηt , the derivatives of the mapping function,Φt can be obtained by

Φt φ ηt

ηt . 5.21

6. Circular Slits

The canonical regionU_cconsists ofmslits along the circle|Φ|R_jwherej 1,2, . . . , mand R1, R2, . . . , Rmare unknown real constants. The boundary values of the mapping functionΦ are given by

Φ ηt

Rt e^iθt, 6.1

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whereθt represents the boundary correspondence function andRt 1, R2, . . . , Rm . By using the same reasoning as inSection 4, we get

Φ ηjt

sign

θt Rjt i T

ηjt Φ ηjt Φ

η_jt , 6.2

ηt Φ ηt Φ

ηt iθt . 6.3

The mapping functionΦz can be uniquely determined by assumingΦβ 0,Φ∞ ∞ andΦ∞ 1. Thus, the mapping functionΦcan be expressed as12

Φz z−β

e^Fz. 6.4

By taking logarithm on both sides of6.4 , we obtain F

ηt

lnRt iθ−log

ηt −β

. 6.5

Hence,6.5 satisfies boundary values inTheorem 2.1withAt 1,

ht lnR₁,lnR₂, . . . ,lnR_m , γt −lnηt −β. 6.6

By obtaininghj, the values ofRjcan be obtained by

R_je^h^j. 6.7

By diﬀerentiating6.5 and comparing with6.3 yields

iθt F ηt

ηt ηt

ηt −β. 6.8

In view ofAηt ,ft Fηt andgt 1/ηt −β wherefz is analytic inΩ⁻and gz is analytic inΩ, we rewrite6.8 as

Af ηt

iθt −Ag ηt

. 6.9

LetAgt ψiϕ. Then by18, Theorems 2c and 2d , we obtain IN^∗

θt −ϕt

Mψt , 6.10

I−N^∗ ϕt Mψt . 6.11

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Subtracting6.11 from6.10 , we get

IN^∗ θt 2ϕt . 6.12

Since the images of the curveΓ1,Γ2, . . . ,Γmare slits, we haveθj2π −θj0 0. Thus

Jθt 0,0, . . . ,0 . 6.13

Hence the unknown functionθt is the unique solution of the integral equation

IN^∗J θt 2ϕt , 6.14

where

ϕt Im At g

ηt Im

ηt

ηt −β . 6.15

By squaring both sides of6.2 and dividing the result byηt −β ², we obtain Φ

ηt 2

ηt −β2 − Rt ² ηt −β2T

ηt ₂ Φ ηt 2

Φ

ηt ². 6.16 Upon comparing 6.16 with 3.2 leads to a choice of Pηt Φηt ²/ηt −β ², P∞ 1,Dηt Φηt ,D∞ 1,ct −Rt ²/ηt −β² . Hence,

φt Φ ηt

ηt 6.17

satisfies the integral equation3.16 with νt − Rt ²

ηt −β₂ηt ηt . 6.18

Numerical evidence shows thatλ−1 is an eigenvalue ofKs, t of multiplicitym, thus one needs to addmconditions. SinceΦz is assumed to be single-valued, it is also required that the unknown mapping functionΦz satisfies4

Jj

φt dt

Γj

Φ η

dη0, j1,2, . . . , m, 6.19

that is,

Jφ0. 6.20

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Then,φt is the unique solution of the integral equation

IKJ φt νt . 6.21

By obtainingφηt , the derivatives of the mapping function,Φt can be obtained by Φt φ

ηt

ηt . 6.22

7. Radial Slits

The canonical regionU_rconsists ofmslits alongmsegments of the rays with argΦ θ_j, j 1,2, . . . , m. Then, the boundary values of the mapping functionΦare given by

Φ ηt

rt e^iθte^Rte^iθt, 7.1 where the boundary correspondence functionθt θ1, θ₂, . . . , θ_m now becomes real constant function andRt is an unknown function. By taking logarithmic derivative on7.1 , we obtain

ηt Φ ηt Φ

ηt Rt . 7.2

It can be shown that the mapping functionΦz can be determined using Φ

ηt

ηt Φ ηt

Rt . 7.3

The mapping functionΦz can be uniquely determined by assumingΦβ 0,Φ∞ ∞ andΦ∞ 1. Thus, the mapping functionΦz can be expressed as12

Φz z−β

e^iFz. 7.4

By taking logarithm on both sides of7.4 and multiplying the result by−i, we obtain F

ηt

θ−iRt ilog

ηt −β

. 7.5

Hence,7.5 satisfies boundary values inTheorem 2.1withAt 1, ht θ1, θ2, . . . , θm , γt −Arg

ηt −β

. 7.6

By obtaininght , one can obtain the values ofθt by

θ_j h_j forj 1,2, . . . , m. 7.7

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Then, by diﬀerentiating7.5 and comparing with7.2 yields

iRt −F ηt

ηt i ηt

ηt −β. 7.8

In view ofA ηt ,ft Fηt andgt i/ηt −β wherefz is analytic inΩ⁻and gz is analytic inΩ, we rewrite7.8 as

Af ηt

iRt −Ag ηt

. 7.9

LetAgt ψiϕ. Then by18, Theorems 2c and 2d , we obtain IN^∗

ϕt −Rt

−Mψt ,

I−N^∗ ϕt Mψt . 7.10

Adding these equations, we get

IN^∗ Rt 2ϕt . 7.11

Since the images of the curveΓ1,Γ2, . . . ,Γmare slits, we haveRj2π −Rj0 0. Therefore JRt 0,0, . . . ,0 . 7.12

Hence, the unknown functionRt is the unique solution of the integral equation

IN^∗J Rt 2ϕt , 7.13

where

ϕt Im At g

ηt Im

iηt

ηt −β . 7.14

The boundary relationship7.3 can be rewritten as Φ

ηt

±e^iθ^jT

ηt Φ

ηt . 7.15 Squaring both sides of7.15 yields

Φ ηt

e^2iθ^jT ηt ₂

Φ ηt

. 7.16

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Upon comparing 7.16 with 3.1 leads to a choice ofPηt 1, P∞ 1, Dηt Φηt ,D∞ 1,ct e^i2θ^j. Hence,

φt Φ ηt

ηt 7.17

νt e^2iθ^j^tηt ηt . 7.18

Numerical evidence shows thatλ −1 is an eigenvalue ofKs, t of multiplicitym, which suggests one needs to addmconditions. SinceΦz is assumed to be single-valued, hence it is also required that the unknown mapping functionΦz satisfies4

Jj

φt dt

Γj

Φ η

dη0, j1,2, . . . , m, 7.19

that is,

Jφ0. 7.20

Then,φt is the solution of the following integral equation

IKJ φt νt . 7.21

By obtainingφηt , the derivatives of the mapping functionΦt can be found using

Φt φ ηt

ηt . 7.22

8. Parallel Slits

The canonical regionUpconsists of amparallel straight slits on thew-plane. LetBe^iπ/2−θ, then the boundary values of the mapping functionΦare given by

BΦ ηt

Rt iδt , 8.1

where θ is the angle of intersection between the lines ReBΦ R_j and the real axis.

Rt R1t , R2t , . . . , RMt is a piecewise real constant function andδt is an unknown function. It can be shown that8.1 can be written as

BΦ ηt

ηt iδt . 8.2

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The mapping function Φz can be uniquely determined by assuming Φ∞ ∞ and lim_{z→ ∞}Φz −z0. Thus, the mapping functionΦcan be expressed as12

Φz zBFz , 8.3

whereFz is an analytic function withF∞ 0. By multiplying both sides of8.3 withB, we obtain

F ηt

BΦ ηt

−Bηt . 8.4

Hence,8.4 satisfies the boundary values inTheorem 2.1withAt 1,

ht R1, R₂, . . . , R_m γt −Bηt . 8.5 Diﬀerentiating8.4 and comparing the result with8.2 yield

iδt Bηt ηt F ηt

. 8.6

In view ofA ηt ,ft Fηt andgt B, wherefz is analytic inΩ⁻ andgz is analytic inΩ, we rewrite8.6 as

Af ηt

iδt −Ag ηt

. 8.7

AssumingAgt ψiϕ, then by18, Theorems 2c and 2d , we obtain IN^∗

δt −ϕt

Mψt ,

I−N^∗ ϕt Mψt . 8.8

These two equations yields

IN^∗ δt 2ϕt . 8.9

Note that, the images of the curvesΓ1,Γ2, . . . ,Γmare slits, so we haveδ_j2π −δj0 0, which implies

Jδt 0,0, . . . ,0 . 8.10

Hence, the unknown functionδt is the unique solution of the integral equation

IN^∗J δt 2ϕt , 8.11

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where

ϕt Im At g

ηt Im!

ηt B"

. 8.12

From8.1 , we introduce an analytic functionϑηt such that

ϑ ηt

e^Fze^−Bηte^{Rt iδt}, 8.13

whereϑ∞ 1. By diﬀerentiating8.13 with respect tot, we obtain

ϑ ηt

ηt

iδt −Bηt ϑ

ηt

. 8.14

Letσt be an analytic function such that it has the following representation

σ ηt

ϑ ηt

Bϑ ηt

, 8.15

where σ∞ B. From 8.13 –8.15 it can be shown that, the functionϑηt can be rewritten as

ϑ ηt

e^R^je⁻^ReBηtsignδt

i T

ηt σ ηt σ

ηt . 8.16 By squaring both sides of8.16 , the signδt is eliminated, that is,

ϑ ηt ₂

−e^2R^je^{−2 ReBηt}T

ηt ₂ σ ηt 2

σηt ². 8.17

Comparing 8.17 with 3.2 leads to a choice of Pz ϑz ², P∞ 1, Dz σz , D∞ B,ct −e^2R^je^{−2 ReBηt}. Hence,

φt σ ηt

ηt 8.18

νt −e^2R^je^{−2 ReBηt}ηt BBηt . 8.19

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Numerical evidence shows thatλ −1 is an eigenvalue ofKs, t of multiplicitym, which suggests one needs to addmconditions. Sincee^Bη^j^tϑηjt |^2π₀ 0, hence we can havem additional conditions for the integral equation above as in the following:

Jj

d dt

e^Bη^j^tϑ_j η_jt

dt0, _2π

0

e^Bη^j^t ϑ_j

η_jt Bϑ

η_jt

ηt dt0, _2π

0

e^Bη^j^t σ

ηjt

ηt dt0, _2π

0

e^Bη^j^tφ η_jt

dt0, q1,2, . . . , m.

8.20

We define Fredholm operator L as

Lμs

Jj

e^Bη^j^tμt dt. 8.21

Then,φt is the solution of the following integral equation:

IKL φt νt . 8.22

Hence, by obtainingφηt , the functionσt can be found by

σt φ ηt

ηt . 8.23

This allows the value for ϑηt to be calculated from 8.16 , which in turn allows the boundary values for the mapping functionΦηt to be calculated by

Φ ηt

ηt Blogϑ ηt

. 8.24

9. Numerical Examples

Since the boundaries Γj are parameterized by ηt which are 2π-periodic functions, the reliable method to solve the integral equations are by means of Nystr ¨om method with trapezoidal rule21. Each boundary will be discretized bynnumber of equidistant points.

The resulting linear systems are then solved by using Gaussian elimination. For numerical examples, we choose regions with connectivities one, two, three and four. For the region

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Table 1: Error norm wj−wj ∞forExample 9.1.

n Uc Ur Up,π/3

32 6.2×10⁻¹² 4.6×10⁻¹⁴ 8.8×10⁻¹⁶

64 9.1×10⁻¹⁴ 9.6×10⁻¹⁵ —

with connectivity one, we compare our result with the analytic solution given in12. All the computations were done by using MATLAB R2008a software.

Example 9.1. Consider an unbounded regionΩ⁻bounded by a unit circle

Γ1t e^−it, 0≤t≤2π . 9.1

We choose the special pointβ2.51.5i. The exact mapping function forU_c, U_r, andU_pare given respectively by12

w_cz−β β−z 1−βz,

w_r z−β−z−β βz ,

wp z e^2iθ z .

9.2

For this example, we compare the error for each boundary value between our method and the exact mapping function. SeeTable 1for Error Norm of||wj−wj||∞.

Example 9.2. Consider an unbounded regionΩ⁻bounded by a circle and an ellipse Γ1t 2ie^−it, 0≤t≤2π ,

Γ2t −2cost−2i sint, 0≤t≤2π . 9.3

Figure 2shows the region and its five canonical images by using our proposed method.

Example 9.3. Consider an unbounded regionΩ⁻bounded by 3 circles Γ1t 2e^−it, 0≤t≤2π ,

Γ2t −1i√

30.5e^−it, 0≤t≤2π , Γ3t −1−i√

31.5e^−it, 0≤t≤2π .

9.4

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−5 −4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1 0 1 2 3 4

−6 −5 −4 −3 −2 −1 0 1 2

−4

−3

−2

−1 0 1 2

−1 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

−4 −3 −2 −1 0 1 2 3 4

−3

−2

−1 0 1 2 3

−6 −4 −2 0 2 4

−5

−4

−3

−2

−1 0 1 2 3 4 5

−4 −3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

−0.5

Figure 2: The original regionΩ⁻and its canonical images withθpπ/3 for parallel slits.

This example has also been considered in6,12.Figure 3shows the regions and its five canonical images by using our proposed method. SeeTable 3for numerical comparison between our parameter values see Table 2 those in 12. Note that our method has considered exterior unit disk with slits as a canonical region while12has considered interior

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−5 0 5

−4

−3

−2

−1 0 1 2 3

−5 −4 −3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

−4 −3 −2 −1 0 1 2 3 4 5

−3

−2

−1 0 1 2 3

−4 −3 −2 −1 0 1 2 3 4 5

−6

−5

−4

−3

−2

−1 0 1 2

−6 −4 −2 0 2 4 6

−6

−5

−4

−3

−2

−1 0 1 2 3 4

Figure 3: The original regionΩ⁻and its canonical images withθpπ/2 for parallel slits.

unit disk with slits. Thus, in computing the error forUd, we need to change the values for U_d to 1/|Φz |. See Table 4for Error Norm of max_1≤j≤3||wj−w_j||_∞. We also compared the condition number of our linear system for eachnwith6,12, see Figures4and5. The results show that for our integral equations for findingΦz that is,4.20 ,5.20 ,6.21 ,7.21 and 8.22 , the condition numbers are almost constant except for5.20 . This is because the kernel

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Table 2: The values for approximated parameters inExample 9.3withn256.

j RjUd RjUa RjUc θjUr RjUp,π/2

1 1.0000000000 1.0000000000 2.6958524041 −0.23582974094 1.32203173492 2 2.9672620504 0.3515929850 2.9121788457 2.24673051228 −0.78705294688 3 2.7249636495 0.1792099292 2.2653736950 −2.00502589294 −0.69725704737

Table 3: Error norm max_1≤j≤3 Rj−Rj ∞of our method with12forExample 9.3.

n Ud Ua Uc Ur Up,π/2

32 2.7×10⁻¹¹ 3.4×10⁻¹¹ 2.9×10⁻⁰¹ 1.6×10⁻⁰¹ 2.0×10⁻¹⁰ 64 5.6×10⁻¹⁷ 1.1×10⁻¹⁶ 4.4×10⁻⁰⁵ 2.0×10⁻⁰⁵ 2.2×10⁻¹⁶ 128 7.8×10⁻¹⁶ 4.7×10⁻¹⁶ 2.2×10⁻⁰⁹ 4.1×10⁻¹⁰ 8.9×10⁻¹⁶ 256 1.6×10⁻¹⁵ 9.99×10⁻¹⁶ 3.8×10⁻¹² 5.2×10⁻¹³ 1.3×10⁻¹⁵

Table 4: Error norm max_1≤j≤3 wj−wj ∞of our method with12forExample 9.3.

n Ud Ua Uc Ur Up,π/2

32 1.1×10⁻⁰⁷ 1.1×10⁻⁰⁷ 0.15 0.16 1.5×10⁻⁰⁶

64 6.1×10⁻¹⁴ 2.9×10⁻¹⁴ 6.2×10⁻⁰⁵ 2.0×10⁻⁰⁵ 1.1×10⁻¹³ 128 7.5×10⁻¹⁴ 2.1×10⁻¹⁴ 2.2×10⁻⁰⁹ 4.2×10⁻¹⁰ 4.4×10⁻¹² 256 2.2×10⁻¹³ 4.7×10⁻¹³ 3.3×10⁻¹² 4.1×10⁻¹² 9.5×10⁻¹²

10² 10¹

10² 10³

n Ud

Ua

Uc

Ur

Up

Condition number

Figure 4: Condition numbers of the matrices for our method forUd, Ua, Uc, UrandUp.

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10² 10²

10⁴ 10⁶ 10⁸

n G.N.K.

Adj. G.N.K

C.S. circular C.S. radial

Condition number

Figure 5: Condition number of the matrices for generalized Neumann kernelG.N.K , adjoint generalized Neumann kernelAdj.G. N. K , charge simulation for circular slitsC.S. circular and charge simulation for radial slitsC.S. radial .

involvesθt , which varies with the number of collocation points,n. However, this does not have any eﬀect on the accuracy of the method.

Example 9.4. Consider an unbounded regionΩ⁻of 4-connectivity with boundaries

Γ1t 32ie^−it, 0≤t≤2π , 9.5 Γ2t −32ie^−it, 0≤t≤2π , 9.6 Γ3t −3−2i0.7 cost−1.4i sint, 0≤t≤2π , 9.7 Γ4t 3−2i0.7 cost−1.4i sint, 0≤t≤2π . 9.8

Figure 6shows the region and its five canonical images by using our proposed method.

10. Conclusion

In this paper, we have constructed a unified method for numerical conformal mapping of unbounded multiply connected regions onto canonical slit regions. The advantage of this method is that the integral equations are all linear which overcomes the nonlinearity problem encountered in 10. From the numerical experiments, we can conclude that our method works on any finite connectivity with high accuracy. By computing the boundary values of the mapping function, the exterior points will be calculated by means of Cauchy’s integral formula.