Numerical conformal mapping and its inverse of ... - Semantic Scholar

Downloaded from http://rspa.royalsocietypublishing.org/ on December 30, 2015

rspa.royalsocietypublishing.org

Research Cite this article: Yunus AAM, Murid AHM, Nasser MMS. 2014 Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions. Proc. R. Soc. A 470: 20130514. http://dx.doi.org/10.1098/rspa.2013.0514 Received: 2 August 2013 Accepted: 4 November 2013

Subject Areas: applied mathematics, computational mathematics, integral equations Keywords: numerical conformal mapping, unbounded multiply connected regions, adjoint generalized Neumann kernel Author for correspondence: A. H. M. Murid e-mail: [email protected]

Dedicated to Prof. M. Zuhair Nashed, in appreciation of a long friendship and lasting contributions.

Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions A. A. M. Yunus1 , A. H. M. Murid2,3 and M. M. S. Nasser4,5 1 Faculty of Science and Technology, Universiti Sains Islam Malaysia,

71800, Bandar Baru Nilai, Negeri Sembilan, Malaysia 2 Department of Mathematical Sciences, Faculty of Science, and 3 UTM Centre of Industrial and Applied Mathematics (UTM-CIAM), Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia 4 Department of Mathematics, Faculty of Science, King Khalid University, PO Box 9004, Abha, Saudi Arabia 5 Department of Mathematics, Faculty of Science, Ibb University, PO Box 70270, Ibb, Yemen This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions. For each canonical region, two integral equations are solved before one can approximate the boundary values of the mapping function. Cauchy’s-type integrals are used for computing the mapping function and its inverse for interior points. This method also works for regions with piecewise smooth boundaries. Three examples are given to illustrate the effectiveness of the proposed method.

1. Introduction Conformal mappings have been a popular technique to solve several problems in the fields of science

2013 The Author(s) Published by the Royal Society. All rights reserved.


Let Ω − be an unbounded multiply connected region of connectivity m. The boundary Γ consists of m Jordan curves Γj , j = 1, 2, . . . , m, i.e. Γ = Γ1 ∪ Γ2 ∪ · · · ∪ Γm . The boundaries Γj are assumed in clockwise orientation (figure 1).

...................................................

2. Notations and auxiliary material

2

rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130514

and engineering. However, exact conformal maps are known for some regions only. One way to overcome this limitation is by means of numerical methods. The Szegö kernel and the Bergman kernel of a simply connected region are well-known reproducing kernels and are connected to the Riemann mapping function which maps a simply connected region onto a unit disc. These kernels can be computed via Fredholm integral equations as shown in [1–3]. Boundary integral equations related to boundary relationships satisfied by functions which are analytic in simply connected regions (bounded and unbounded) or bounded doubly connected regions with smooth Jordan boundaries have been given by [4–6]. Special realizations of these integral equations are the integral equations related to the Szegö kernel, Bergmann kernel, Riemann map, exterior mapping and Ahlfors map. The kernels of these integral equations are the Neumann kernel and the Kerzman–Stein kernel. Extensions of this approach to conformal mappings of bounded and unbounded multiply connected regions onto some canonical regions are given in [7–12]. For some other approaches of conformal mappings of multiply connected regions, e.g. [13–31]. Various applications of conformal mappings in science and engineering are considered in, e.g., [22,32–35]. There exist many canonical regions with regards to conformal mapping of multiply connected regions, e.g. [18,22,36–39]. Koebe [36] gives an example of 39 types of canonical regions. However, most of the works mentioned above are for the first type of Koebe’s canonical slit region as illustrated in [36, figs 1–5]. In this paper, we present a new unified method for univalent conformal mapping of unbounded finitely connected regions and its inverse onto several classes of canonical regions via a boundary integral equation method with the adjoint generalized Neumann kernel. We consider the first 13 canonical regions shown in [36, figs 1–13]. Nasser [37] has applied the boundary integral equation method for numerical conformal mapping onto these canonical regions by reformulating the conformal mapping problem as a Riemann–Hilbert (RH) problem. Then, integral equations with the generalized Neumann kernel are constructed to solve the RH problem. Nasser [37] is an extension of the author’s two previous papers [23,24]. This method only works for solving the direct mapping problem. Recently, a fast boundary integral equation for numerical conformal mapping onto a strip with rectilinear slit has been given in [40]. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the Fast Multipole Method. In this paper, by computing the mapping function via the adjoint generalized Neumann kernel, we are able to compute its inverse map and the time taken for Matlab to run the program also reduces. Amano & Okano [15] and DeLillo et al. [41] have also developed their own techniques for numerical conformal mapping onto circular and radial slit regions where Amano & Okano [15] use charge simulation methods while DeLillo et al. [41] use an explicit formula from an unbounded circular region and later solve it using the least-squares method. In this paper, we construct two integral equations which can be used to compute the mapping function and its inverse from any unbounded multiply connected regions onto Koebe’s first 13 canonical regions. One integral equation is used to find the parameters of the canonical regions. The other integral equation is used to compute the derivative of the boundary correspondence function. The plan of this paper is as follows: §2 presents some auxiliary materials. Section 3 presents a boundary integral equation with the adjoint generalized Neumann kernel. In §§4–7, we present the derivation for numerical conformal mapping of unbounded multiply connected regions onto canonical regions and its inverse. In §8, we give some examples to illustrate the effectiveness of our method. Finally, §9 presents a short conclusion.


G2

Gm

3 ...................................................


G1 W–

Figure 1. An unbounded multiply connected region Ω − with connectivity m.

U1

U2

U4

U3

Figure 2. The classes of the canonical region. (Online version in colour.)

This paper has introduced slightly different notations from our previous works [12,23,24,26, 31,37,42–44] so as to give a detailed explanation of the presented method. The curves Γj are parametrized by 2π -periodic twice continuously differentiable complexvalued function ηj (t), t ∈ [0, 2π ], j = 1, 2, . . . , m, ⎫ Γ1 : η1 (t), t ∈ [0, 2π ] ⎪ ⎪ ⎪ ⎪ ⎬ .. (2.1) . ⎪ ⎪ ⎪ ⎪ and Γm : ηm (t), t ∈ [0, 2π ],⎭ with non-vanishing first derivatives, i.e. ηj (t) =

dηj (t) dt

= 0,

t ∈ [0, 2π ],

j = 1, . . . , m.

(2.2)

In this paper, we shall consider four canonical regions which are annulus with spiral slits (U1 ), unit disc with spiral slits (U2 ), spiral slits region (U3 ) and rectilinear slits region (U4 ) (figure 2). These canonical regions are the same as the first 13 canonical regions shown in [36, figs 1–13]. Let Φ(z) be the conformal mapping function that maps Ω − onto any of the canonical regions mentioned above, z1 is a prescribed point inside Γ1 , z2 is a prescribed point inside Γ2 and α is a prescribed point located in Ω − . In this paper, we determine the mapping function Φ(z) by computing two real functions which are an unknown function Sj (t) and a piecewise constant real function Rj .

3. Adjoint generalized Neumann kernel Let A1,j (t) and A2,j (t) be complex-valued continuously differentiable 2π -periodic functions for all t ∈ [0, 2π ] and j = 1, . . . , m. The adjoint function A˜ p,j (t) of Ap,j (t) is defined by A˜ p,j (t) =

ηj (t) Ap,j (t)

.

(3.1)


Throughout this paper, we shall assume the functions A1,j (t) and A2,j (t) are constant functions, so we shall eliminate the variable t, i.e. we shall define (3.2)

where θj are given real constants for j = 1, . . . , m. Hence, the adjoint functions A˜ 1,j (t) and A˜ 2,j (t) are given by A˜ 1,j (t) = A2,j ηj (t) and

A˜ 2,j (t) = A1,j ηj (t).

(3.3)

For p = 1, 2, we consider two real kernels formed with Ap,j as [44] ηj (t) Ap,l 1 Np,l,j (s, t) = Im π Ap,j ηj (t) − ηl (s) and

ηj (t) Ap,l 1 Mp,l,j (s, t) = Re , π Ap,j ηj (t) − ηl (s)

where s, t ∈ [0, 2π ] and j, l = 1, 2, . . . , m. The kernel Np,l,j (s, t) is known as the generalized Neumann kernel formed with Ap,j and ηj (t). For j = l, all kernels Np,l,j (s, t) and Mp,l,j (s, t) are continuous on [0, 2π ] × [0, 2π ] because ηj (t) = ηl (s) for all s, t ∈ [0, 2π ]. When j = l, the kernel Np,j,j (s, t) is continuous and takes on the diagonal the values ηj (t) 1 Im Np,j,j (t, t) = . 2π ηj (t) The kernel Mp,j,j (s, t) has a cotangent singularity where it can be written as Mp,j,j (s, t) = −

s−t 1 cot + M1,p,l,j (s, t), 2π 2

with a continuous kernel M1,p,l,j (s, t) which takes on the diagonal the values ηj (t) 1 Re M1,p,j,j (t, t) = . 2π ηj (t) ˜ p,l,j formed with the adjoint ˜ p,l,j and the real kernel M The generalized Neumann kernel N function A˜ p,l (t) and ηj (t) are defined by ηj (t) A˜ p,l (s) 1 ˜ Np,l,j (s, t) = Im π A˜ p,j (t) ηj (t) − ηl (s)

ηj (t) A˜ p,l (s) 1 ˜ and Mp,l,j (s, t) = Re . π A˜ p,j (t) ηj (t) − ηl (s)

Then, ˜ p,l,j (s, t) = −N∗ (s, t) N p,l,j

˜ p,l,j (s, t) = −M∗ (s, t), and M p,l,j

(3.4)

∗ (s, t) = N where Np,l,j p,j,l (t, s) is the adjoint kernel of the generalized Neumann kernel Np,l,j (s, t) and ∗ Mp,l,j (s, t) = Mp,j,l (t, s) is the adjoint kernel of the kernel Mp,l,j (s, t). It is known that λ = 1 is not an eigenvalue of the kernel Np,l,j and λ = −1 is an eigenvalue of the kernel Np,l,j with multiplicity m. The eigenfunctions of Np,l,j corresponding to the eigenvalue λ = −1 are {χ [1] , χ [2] , . . . , χ [m] }, where 1, ζ ∈ Γj , [j] χ (ζ ) = 0, elsewhere,

j = 1, 2, . . . , m (see [44] for details).

...................................................

and A2,j = e−i(π/2−θj ) ,


A1,j = ei(π/2−θj )

4


For l, j = 1, 2, . . . , m, we define a real kernel Jl,j (s, t), where s, t ∈ [0, 2π ] by

5

Let H be the space of all real Hölder continuous 2π -periodic function and L is the subspace of H which contains the piecewise real constant functions. Hence, we have the following theorems from [31,43]. Theorem 3.1. For any given function γ ∈ H, the m × m system of integral equations μj (t) +

m 2π l=1

0

∗ [Np,j,l (t, s) + Jj,l (t, s)]μl (s) ds = γ (ηj (t)),

j = 1, 2, . . . , m,

(3.6)

is uniquely solvable. [k]

[k]

[k]

Theorem 3.2. For fixed integers p = 1, 2 and k = 1, 2, . . . , m, let {ϑ1 , ϑ2 , . . . , ϑm } be the unique solution of the m × m system of integral equations [k]

ϑj (t) +

m 2π l=1

0

∗ [Np,j,l (t, s) + Jj,l (t, s)]ϑl (s) ds = −χ [k] (ηj (t)), [k]

j = 1, 2, . . . , m.

(3.7)

Let γ , μ ∈ H and h, ν ∈ L such that Ap,j F(ηj (t)) = γj (t) + hj + i[μj (t) + νj ],

j = 1, 2, . . . , m

(3.8)

are boundary values of a function F(z) analytic in Ω − with F(∞) = 0. Then the elements of the piecewise constant functions h = (h1 , h2 , . . . , hm ) and ν = (ν1 , ν2 , . . . , νm ) are given by hk =

m 1 2π [k] γj (t)ϑj (t) dt 2π 0

(3.9)

m 1 2π [k] μj (t)ϑj (t) dt. 2π 0

(3.10)

j=1

and νk =

j=1

4. An annulus with spiral slits region Consider the class of canonical region U1 , that is an annulus centred at the origin with m − 2 logarithmic spiral slits. We assume that Φ(z) maps the curve Γ1 onto a unit circle |Φ| = 1, the curve Γ2 onto the circle |Φ(z)| = R2 and the curves Γj , j = 3, 4, . . . , m onto slits on the logarithmic spiral [39] Im(e−iθj log Φ(ηj (t))) = Rj ,

t ∈ [0, 2π ],

(4.1)

where R2 , . . . , Rm are undetermined real constants. The parameters θj , j = 3, 4, . . . , m are not predetermined by Ω − , hence they can be given any real constants such that each θj represents the angle of intersection between the logarithmic spiral and any ray emanating from the origin. Note that the slits are always traversed twice. For θ = π/2, the logarithmic spiral slit is a circular

...................................................

(3.5)


⎧ ⎨ 1 , if l = j, Jl,j (s, t) = 2π ⎩0, if l = j,


slit centred at the origin, while for θ = 0 the spiral is a radial slit pointing at the origin. We choose θ1 = θ2 = π/2. Then, the boundary values of the mapping function Φ(z) satisfy (4.2)

where Sj (t) is unknown real-valued function and ⎧ ⎪ j = 1, ⎪ ⎨0, ˆRj = ln R2 , j = 2, ⎪ ⎪ ⎩−R , j = 3, . . . , m. j

(4.3)

The mapping function Φ(z) can be uniquely determined by assuming that Φ(∞) > 0. Thus, the mapping function can be expressed as [37]

Φ(z) = c

z − z2 z − z1

eF(z) ,

(4.4)

where F(z) is an analytic function with F(∞) = 0 and c = Φ(∞) is an undetermined real constant. By taking the logarithm of both sides of (4.4), we obtain log Φ(ηj (t)) = ln c + log

ηj (t) − z2

j = 1, 2, . . . , m.

+ F(ηj (t)),

ηj (t) − z1

(4.5)

Then by multiplying (4.5) with A1,j and applying (4.2), we get A1,j F(ηj (t)) = Rˆ j + iSj (t) − A1,j ln c − A1,j log

ηj (t) − z2

ηj (t) − z1

,

= Rˆ j + iSj (t) − ln c(sin θj + i cos θj ) + γj (t) + iμˆ j (t), j = 1, 2, . . . , m, where

γj (t) + iμˆ j (t) = −A1,j log

ηj (t) − z2

(4.6)

ηj (t) − z1

.

By differentiating (4.6) with respect to t, we get A1,j F (ηj (t))ηj (t) = γj (t) + i(Sj (t) + μˆ j (t)),

j = 1, 2, . . . , m,

(4.7)

where γj (t) + iμˆ j (t) = −A1,j ηj (t)

z2 − z1 . (ηj (t) − z2 )(ηj (t) − z1 )

In view of A˜ 2,j = A1,j ηj (t) and f (z) = F (z) + (z2 − z1 )/((z − z2 )(z − z1 )) is analytic in Ω − , we rewrite (4.7) as A˜ 2,j f (ηj (t)) = iSj (t),

j = 1, 2, . . . , m.

Then by [45, Theorem 1], we obtain Sj (t) +

m 2π l=1

0

∗ N2,j,l (t, s)Sl (s) ds = 0,

j = 1, 2, . . . , m

(4.8)

for t ∈ [0, 2π ]. However, this integral equation is not uniquely solvable [44, Theorem 12]. Note that, the image of the curve Γ1 is anticlockwise oriented, the image of the curve Γ2 is clockwise

...................................................

j = 1, 2, . . . , m,


A1,j log Φ(ηj (t)) = Rˆ j + iSj (t),

6


oriented, and the images of the curves Γj , j = 3, 4, . . . , m are slits that are traversed twice. So, we have S1 (2π ) − S1 (0) = 2π , S2 (2π ) − S2 (0) = −2π , Sj (2π ) − Sj (0) = 0 for j = 3, . . . , m. Hence,

where

0

Jj,l (t, s)Sl (s) ds = h˜ j ,

j = 1, 2, . . . , m,

(4.9)

⎧ ⎪ j = 1, ⎪ ⎨1, ˜hj = −1, j = 2, ⎪ ⎪ ⎩0, j = 3, 4, . . . , m.

(4.10)

By adding (4.9) to (4.8), we conclude that the unknown functions S1 (t), . . . , Sm (t) satisfy the following system integral equations: Sj (t) +

m 2π l=1

0

∗ [N2,j,l (t, s) + Jj,l (t, s)]Sl (s) ds = h˜ j ,

t ∈ [0, 2π ],

j = 1, . . . , m,

(4.11)

which in view of theorem 3.1 is uniquely solvable. The function Sj (t) can be computed as an antiderivative of its derivative Sj (t). The function Sj (t) is 2π -periodic. Thus, it can be represented by a Fourier series [j]

Sj (t) = a0 +

∞

[j]

ak cos kt +

k=1 [j]

[j]

∞

[j]

bk sin kt.

(4.12)

k=1

[j]

The values of a0 , ak and bk are computed by using Matlab’s function ‘fft’. Hence, the function Sj (t) can be written as Sj (t) = Sj (t) dt + νˆ j = ρj (t) + νˆ j ,

j = 1, 2, . . . , m,

(4.13)

where the function ρj (t) can be calculated by Fourier series representation as ρj (t) =

∞ [j] ak [j] Sj (t) dt = a0 t + k k=1

sin kt −

∞ [j] bk k=1

k

cos kt

(4.14)

and νˆ j is undetermined real integration constant and should be calculated. The boundary values of the function F in (4.6) can be written as A1,j F(ηj (t)) = hj + γj (t) + i(μj (t) + νj ),

j = 1, 2, . . . , m,

(4.15)

νj = νˆ j − cos θj ln c.

(4.16)

where μj (t) = ρj (t) + μˆ j (t),

ˆ j − sin θj ln c, hj = R

Note that the functions γj (t) and μj (t) in (4.15) are known, and the function F satisfies the assumptions of theorem 3.2. Hence, the constants hj and νj can be computed from theorem 3.2. ˆ 3 , . . . , Rˆ m and νˆ 1 , νˆ 2 , . . . , νˆ m can be computed from (4.16). Then the Hence, the constants c, Rˆ 2 , R parameters R2 , R3 , . . . , Rm can be computed from (4.3) and the values of the function Sj (t) can be computed from (4.13). By obtaining all the information above, the mapping function at the boundary points can be calculated from (4.2) as ˆ

Φ(ηj (t)) = eA2,j (Rj +iSj (t)) ,

j = 1, 2, . . . , m.

...................................................

l=1


m 2π

7


Then for all z ∈ Ω − , by Cauchy’s integral formula [46] we have

0

Φ(ηj (t)) ηj (t) − z

ηj (t) dt.

(4.17)

For computing the inverse map, note that the mapping function Φ −1 (w) = z is analytic in the region U1 with a simple pole at w = c. Let G(w) be an analytic function in U1 defined as G(w) = (w − c)Φ −1 (w). Hence, G(w) =

1 2π i

∂U1

(ζ − c)Φ −1 (ζ ) dζ . ζ −w

By introducing ζj (t) = Φ(ηj (t)), then by Cauchy’s integral formula, we have (w − c)Φ −1 (w) =

m

2π

j=1

0

1 2π i

(Φ(ηj (t)) − c)Φ −1 (Φ(ηj (t))) Φ(ηj (t)) − w

A2,j iSj (t)Φ(ηj (t)) dt,

which implies Φ −1 (w) =

m 2π (Φ(ηj (t)) − c)ηj (t) 1 A2,j Sj (t)Φ(ηj (t)) dt. (w − c)2π Φ(ηj (t)) − w 0 j=1

5. The unit disc with spiral slits region Consider the class of canonical region U2 , i.e. the unit disc with m − 1 logarithmic spiral slits. We assume that Φ(z) maps the curve Γ1 onto the unit circle |Φ| = 1 and the curves Γj , j = 2, 3, . . . , m onto slits on the logarithmic spirals [39] Im(e−iθj log Φ(ηj (t))) = Rj ,

(5.1)

where R2 , . . . , Rm are undetermined real constants. The real constants θj , j = 2, 3, . . . , m, have the same geometrical meaning as in §4. We choose θ1 = π/2. Then, as in §4, the boundary values of the mapping function Φ(z) satisfy A1,j log Φ(ηj (t)) = Rˆ j + iSj (t),

j = 1, 2, . . . , m,

(5.2)

where Sj (t) is unknown real-valued function and ˆ j = 0, R −Rj

j = 1, j = 2, 3, . . . , m.

(5.3)

The mapping function Φ(z) can be uniquely determined by assuming Φ(∞) = 0 and limz→∞ zΦ(z) > 0. Thus, the mapping function can be expressed as [37] Φ(z) =

c eF(z) , z − z1

(5.4)

where F(z) is an analytic function in Ω − with F(∞) = 0 and c = limz→∞ zΦ(z) is an undetermined positive real constant.

...................................................

j=1

2π


1 2π i m

w = Φ(z) = c +

8


By taking the logarithm of both sides of (5.4), we obtain

9 j = 1, 2, . . . , m.

(5.5)

A1,j F(ηj (t)) = Rˆ j + iSj (t) − A1,j ln c + A1,j log(ηj (t) − z1 ), = Rˆ j + iSj (t) − ln c(sin θj + i cos θj ) + γj (t) + iμˆ j (t),

j = 1, 2, . . . , m,

(5.6)

where γj (t) + iμˆ j (t) = A1,j log(ηj (t) − z1 ). By differentiating (5.6) with respect to t, we obtain A1,j F (ηj (t))ηj (t) = γj (t) + i(Sj (t) + μˆ j (t)), where γj (t) + iμˆ j (t) = A1,j

ηj (t) ηj (t) − z1

j = 1, 2, . . . , m,

(5.7)

.

As A˜ 2,j (t) = A1,j ηj (t) and f (z) = F (z) − 1/(z − z1 ) is analytic in Ω − , we rewrite (5.7) as A˜ 2,j (t)f (ηj (t)) = iSj (t),

j = 1, 2, . . . , m.

Then by [45, Theorem 1], we get Sj (t) +

m 2π l=1

0

∗ N2,j,l (t, s)Sl (s) ds = 0,

t ∈ [0, 2π ],

j = 1, 2, . . . , m.

(5.8)

This integral equation is not uniquely solvable [44]. Note that, the image of the curve Γ1 is anticlockwise oriented and the images of the curves Γj , j = 2, 3, . . . , m are slits that traversed twice. So we have S1 (2π ) − S1 (0) = 2π and Sj (2π ) − Sj (0) = 0 for j = 2, 3, . . . , m, which implies that m 2π l=1

0

where


j = 1, 2, . . . , m,

(5.9)

1, j = 1, h˜ j = 0, j = 2, 3, . . . , m.

(5.10)

Hence, the unknown function Sj (t) is the unique solution of the system of following integral equations: Sj (t) +

m 2π l=1

0

∗ [N2,j,l (t, s) + Jj,l (t, s)]Sl (s) ds = h˜ j ,

t ∈ [0, 2π ],

j = 1, 2, . . . , m.

(5.11)

For j = 1, 2, . . . , m, the function Sj (t) can be written as Sj (t) = Sj (t) dt + νˆ j = ρj (t) + νˆ j (t). Hence, the boundary values of the function F in (5.6) can be written as A1,j F(ηj (t)) = hj + γj (t) + i(μj (t) + νj ), where μj (t) = ρj (t) + μˆ j (t),

ˆ j − sin θj ln c hj = R

and νj = νˆ j − cos θj ln c.

(5.12)

...................................................

Multiplying (5.5) with A1,j and applying (5.2), we get


log Φ(ηj (t)) = ln c − log(ηj (t) − z1 ) + F(ηj (t)),


j = 1, 2, . . . , m.

Then for all z ∈ Ω − , by Cauchy’s integral formula [46], we have m 1 2π Φ(ηj (t)) w = Φ(z) = ηj (t) dt. 2π i 0 ηj (t) − z

(5.13)

j=1

For computing the inverse map, note that the mapping function Φ −1 (w) = z is analytic in the region U2 with a simple pole at w = 0. Let G(w) be an analytic function in U2 defined as G(w) = wΦ −1 (w). For computing the inverse map, note that the mapping function Φ −1 (w) = z is analytic in the region U2 with a simple pole at w = 0. Let G(w) be an analytic function in U2 defined as G(w) = wΦ −1 (w). Then, by using the same reasoning as in §4, we get m 1 2π Φ(ηj (t))ηj (t) A2,j Sj (t)Φ(ηj (t)) dt. Φ −1 (w) = 2π w 0 Φ(ηj (t)) − w j=1

6. Spiral slits region Consider the canonical region U3 , i.e. the m logarithmic spirals on the entire w-plane, where the logarithmic spiral have the following representation [39] Im(e−iθj log Φ(ηj (t))) = Rj ,

j = 1, 2, . . . , m,

(6.1)

where R1 , . . . , Rm are real constants. The parameters θj , j = 1, 2, . . . , m are given real constants and have the same geometrical meaning as in §4. The boundary values of the mapping function Φ(z) can be written as ˆ j + iSj (t), (6.2) A1,j log Φ(ηj (t)) = R where Sj (t) is unknown real-valued function and Rˆ j = −Rj

j = 1, 2, . . . , m.

(6.3)

The mapping function Φ(z) can be uniquely determined by assuming Φ(α) = 0, Φ(∞) = ∞ and Φ(z) has the Laurent series expansion in a neighbourhood of z = ∞ as [39, p. 112] Φ(z) = z + a0 +

a2 a1 + 2 + ··· . z z

The mapping function can be expressed as [37] Φ(z) = (z − α) eF(z) ,

(6.4)

where F(z) is an analytic in Ω − with F(∞) = 0. Using the same procedure as in §4, we obtain ˆ j + iSj (t) − A1,j log(ηj (t) − α) A1,j F(ηj (t)) = R ˆ j (t) + iSj (t) + γj (t) + iμˆ j (t), =R

j = 1, 2, . . . , m,

(6.5)

where γj (t) + iμˆ j (t) = −A1,j log(ηj (t) − α). Differentiating (6.5) with respect to t, we obtain A1,j F (ηj (t))ηj (t) = γj (t) + i(Sj (t) + μˆ j (t)), where γj (t) + iμˆ j (t) = A1,j

ηj (t) α − ηj (t)

j = 1, 2, . . . , m,

.

(6.6)

...................................................

ˆ

Φ(ηj (t)) = eA2,j (Rj +iSj (t)) ,

10


ˆ j in (5.2) can be obtained by using the same procedure as in the Then, the values of Sj (t) and R previous section. By obtaining all the information above, the mapping function at the boundary points can be calculated by


As A˜ 2,j (t) = A1,j ηj (t), f (z) = F (z) is analytic in Ω − with f (∞) = 0 and g(z) = 1/(α − z) is analytic in

j = 1, 2, . . . , m.

Then by [44, Theorem 2(c)] and [45], we get Sj (t) + μˆ j (t) +

m 2π l=1

0

∗ N2,j,l (t, s)(Sl (s) + μˆ l (s)) ds =

m 2π l=1

0

M∗2,j,l (t, s)γl (s) ds.

(6.7)

We have also A˜ 2,j (t)gj (t) = γj (t) + iμˆ j (t),

j = 1, 2, . . . , m.

Then by [44, Theorem 2(d)] and [45], we get μˆ j (t) −

m 2π l=1

0

∗ N2,j,l (t, s)μˆ l (s) ds = −

m 2π 0

l=1


(6.8)

Then, by adding (6.7) and (6.8), we have Sj (t) +

m 2π l=1

0

∗ N2,j,l (t, s)Sl (s) ds = −2Im

A˜ 2,j (t)

α − ηj (t)

,

t ∈ [0, 2π ],

j = 1, 2, . . . , m.

(6.9)

Integral equation (6.9) is not uniquely solvable [44, Theorem 12]. Note that the images of the curves Γ1 , Γ2 , . . . , Γm are slits that are traversed twice. So we have Sj (2π ) − Sj (0) = 0, which implies m 2π l=1

0

Jj,l (t, s)Sl (s) ds = 0,

t ∈ [0, 2π ],

j = 1, 2, . . . , m.

(6.10)

Hence, the unknown function Sj (t) is the unique solution of the following system of integral equations m 2π A˜ 2,j (t) ∗ Sj (t) + [N2,j,l (t, s) + Jj,l (t, s)]Sl (s) ds = −2 Im (6.11) α − ηj (t) 0 l=1

for t ∈ [0, 2π ], j = 1, 2, . . . , m. For j = 1, 2, . . . , m, the function Sj (t) can be written as Sj (t) = Sj (t) dt + νˆ j = ρj (t) + νˆ j (t). Hence, the boundary values of the function F in (6.5) can be written as A1,j F(ηj (t)) = hj + γj (t) + i(μj (t) + νj ),

(6.12)

where μj (t) = ρj (t) + μˆ j (t),

ˆj hj = R

and νj = νˆ j .

ˆ j in (6.2) can be obtained by using the same procedure as in §4. Then, the values of Sj (t) and R

...................................................

A˜ 2,j (t)f (ηj (t)) = γj (t) + i(Sj (t) + μˆ j (t)),

11


Ω + (the complement of Ω − with respect to the extended complex plane C ∪ {∞}), we rewrite (6.6) as


The mapping function at the boundary points can be calculated by ˆ j +iSj (t)) A2,j (R

j = 1, 2, . . . , m.

,

K(z) =

Φ(z) , z−α

where lim K(z) = 1. z→∞

Then, by Cauchy’s integral formula [46], we have w=z−α +

m Φ(ηj (t)) z − α 2π ηj (t) dt. 2π i 0 (ηj (t) − α)(ηj (t) − z)

(6.13)

j=1

For computing the inverse map, the inverse of the Laurent series expansion for Φ(z) near ∞ has the following representation [39, p. 114] Φ −1 (w) = w + b0 +

b2 b1 + 2 + ··· . w w

Let G(w) be analytic in U3 defined by G(w) =

Φ −1 (w) − α , w

where lim G(w) = 1. w→∞

Then, by Cauchy’s integral formula, we have G(w) = G(∞) +

1 2π i

∂U3

G(ζ ) dζ . ζ −w

By introducing ζj (t) = Φ(ηj (t)), we have z=w+α +

m ηj (t) − α w 2π A2,j Sj (t)Φ(ηj (t)) dt. 2π 0 Φ(ηj (t))(Φ(ηj (t)) − w) j=1

7. Rectilinear slits region Consider the canonical region U4 , that is the entire complex plane with m rectilinear slits on the straight lines [39] Im(e−iθj Φ) = Rj ,

j = 1, 2, . . . , m,

(7.1)

where R1 , . . . , Rm are undetermined real constants and θj , j = 1, 2, . . . , m are the given angles of intersections between the straight lines (7.1) and the real axis. The boundary values of the mapping function Φ(z) can be written as A1,j Φ(ηj (t)) = Rˆ j + iSj (t),

j = 1, 2, . . . , m.

(7.2)

The mapping function Φ(z) can be uniquely determined by assuming Φ(∞) = ∞ and limz→∞ (z − Φ(z)) = 0. Thus, the mapping function can be expressed as [37] Φ(z) = z + F(z),

(7.3)

where F(z) is analytic in Ω − with F(∞) = 0. By multiplying (7.3) with A1,j and applying (7.2), we get A1,j F(ηj (t)) = Rˆ j + iSj (t) − A1,j ηj (t) = Rˆ j + iSj (t) + γj (t) + iμˆ j (t), where γj (t) + iμˆ j (t) = −A1,j ηj (t).

j = 1, 2, . . . , m,

(7.4)

...................................................

For computing the mapping of exterior point, let K(z) be an analytic function for z ∈ Ω − defined as


Φ(ηj (t)) = e

12


Then, by differentiating (7.4) with respect to t, we have

13 j = 1, 2, . . . , m,

(7.5)

where γj (t) + iμˆ j (t) = −A1,j ηj (t). As A˜ 2,j (t) = A1,j ηj (t), f (z) = F (z) is analytic in Ω − with f (∞) = 0 and g(z) = −1 is analytic in Ω + , we rewrite (7.5) as A˜ 2,j (t)f (ηj (t)) = γj (t) + i(Sj (t) + μˆ j (t)),

j = 1, 2, . . . , m.

Then by [44, Theorem 2(c)] and [45], for t ∈ [0, 2π ], j = 1, 2, . . . , m we get Sj (t) + μˆ j (t) +

m 2π 0

l=1

∗ N2,j,l (t, s)(Sl (s) + μˆ l (s)) ds =

m 2π l=1

0


(7.6)

As A˜ 2,j (t)g(ηj (t)) = γj (t) + iμˆ j (t),

j = 1, 2, . . . , m,

by [44, Theorem 2(d)] and [45], we have μˆ j (t) −

m 2π 0

l=1

∗ N2,j,l (t, s)μˆ l (s) ds = −

m 2π l=1

0


(7.7)

Adding (7.6) with (7.7), yields Sj (t) +

m 2π l=1

0

∗ N2,j,l (t, s)Sl (s) ds = 2 Im[A1,j ηj (t)],

t ∈ [0, 2π ],

j = 1, 2, . . . , m.

(7.8)

This integral equation is not uniquely solvable. Note that, the images of the curve Γ1 , Γ2 , . . . , Γm are slits that traversed twice. So we have Sj (2π ) − Sj (0) = 0, which implies 2π 0

Jj,l (t, s)Sl (s) ds = 0,

j = 1, 2, . . . , m.

(7.9)

Hence, the unknown function Sj (t) is the unique solution of the following integral equation: Sj (t) +

m 2π l=1

0

∗ [N2,j,l (t, s) + Jj,l (t, s)]Sl (s) ds = 2 Im[A1,j ηj (t)],

t ∈ [0, 2π ],

(7.10)

j = 1, 2, . . . , m. For j = 1, 2, . . . , m, the function Sj (t) can be written as Sj (t) = Sj (t) dt + νˆ j = ρj (t) + νˆ j (t). Hence, the boundary values of the function F in (7.4) can be written as A1,j F(ηj (t)) = hj + γj (t) + i(μj (t) + νj ), where μj (t) = ρj (t) + μˆ j (t),

ˆ j, hj = R

νj = νˆ j

and j = 1, 2, . . . , m.

(7.11)

...................................................

(ηj (t))ηj (t) = γj (t) + i(Sj (t) + μˆ j (t)),


A1,j F


Let K(z) be an analytic function for z ∈ Ω − defined as K(z) = Φ(z) − z,

where lim K(z) = 0. z→∞

Then by Cauchy’s integral formula [46], we have m 1 2π Φ(ηj (t)) − ηj (t) ηj (t) dt. w=z+ 2π i ηj (t) − z 0

(7.12)

j=1

For computing the inverse map, the inverse of the Laurent series expansion for Φ(z) near ∞ has the following representation [39, p. 114] Φ −1 (w) = w +

b1 b3 b2 + 2 + 3 + ··· . w w w

Let G(w) be an analytic function in U4 defined as G(w) = Φ −1 (w) − w, Then, G(w) =

1 2π i

where lim G(w) = 0. w→∞

∂U4

Φ −1 (ζ ) − ζ dζ . ζ −w

By introducing ζj (t) = Φ(ηj (t)), then by the Cauchy’s integral formula, we have z=w+

m 1 2π ηj (t) − Φ(ηj (t)) A2,j Sj (t) dt. 2π Φ(ηj (t)) − w 0 j=1

8. Numerical examples As the boundaries Γj are parametrized by ηj (t) which are 2π -periodic function, the reliable method to solve the integral equations are by means of Nyström method with trapezoidal rule [47]. Each boundary will be discretized by n number of equidistant points. The resulting linear system is then solved by using Gaussian elimination method. The integral equations presented in §§4–8 can be also used to compute the conformal mapping for the region that contains corner points. However, the integral equation needs to be modified slightly when ηj (t) is a corner point (see [42] for more details). For this case, the integral equations will be solved using the method presented in [48] (see also [42]). Suppose that each boundary component Γj contains pj ≥ 1 corner points located at 2kπ/n, k = 0, 1, . . . , pj − 1. Suppose j (t) be a function which is bijective, strictly monotonically increasing and infinitely differentiable and is defined by [48] (t) = 2π and

v(t) =

1 1 − 3 2

[v(t)]3 [v(t)]3 + [v(2π − t)]3

π −t π

3

+

1 1t−π + , 3 π 2

(8.1)

t ∈ [0, 2π ],

where the functions j (t) and vj (t) satisfy (0) = (2π ) = 0,

v(0) = 0,

v(2π ) = 1.

(8.2)

...................................................

For calculating the mapping of exterior points, note that Φ(z) has the Laurent expansion series near ∞ as [39, p. 104]: a3 a1 a2 Φ(z) = z + + 2 + 3 + ··· . z z z

14


Then the values of Sj (t) and Rˆ j in (7.2) can be obtained by using the same procedure as in §4. By obtaining all the information above, the mapping function at the boundary points can be calculated by ˆ j + iSj (t)), j = 1, 2, . . . , m. Φ(ηj (t)) = A2,j (R


Then a function δj (t) which satisfies 2kπ p

=0

can be defined by ⎧ ⎪ 1 ⎪ ⎪ ⎪ (pj t), ⎪ ⎪ p j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2π ⎪ ⎪ ⎪ ⎪ p (pj t − 2π ) + p , ⎪ ⎪ j j ⎪ ⎨ .. δj (t) = . ⎪ ⎪ ⎪ ⎪ 2(pj − 2)π ⎪ 1 ⎪ ⎪ , ⎪ ⎪ pj (pj t − 2(pj − 2)π ) + pj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2(pj − 1)π 1 ⎪ ⎪ , ⎪ ⎩ p (pj t − 2(pj − 1)π ) + pj j

2π t ∈ 0, , pj 2π 4π t∈ , , pj pj

2(pj − 2)π 2(pj − 1)π t∈ , pj pj 2(pj − 1)π t∈ , 2π , pj

,

For j = 1, 2, . . . , m, we define a new parametrization ζj (t), t ∈ [0, 2π ], of the boundary component Γj by ζj (t) = ηj (δj (t)),

(8.3)

i.e. the boundary Γ will be re-parametrized by Γ1 : ζ1 (t), .. . and Γm : ζm (t),

⎫ ⎪ t ∈ [0, 2π ], ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ t ∈ [0, 2π ].⎭

(8.4)

Let Jj∗ be the parametrization interval [0, 2π ] minus the points at which δj (t) = 0 for j = 1, . . . , m. Then δj (t) = 0

for all t ∈ Jj∗ .

We have ζj (t) = ηj (δj (t))δj (t),

t ∈ [0, 2π ],

j = 1, 2, . . . , m.

(8.5)

Thus, ζj (t) = 0 at each corner point and ζj (t) = 0 for all t ∈ Jj∗ . By introducing the new parametrization ζj (t) of the boundary Γ , a modification of the integral equations (4.11), (5.11), (6.11), (7.10) and (3.7) in theorem 3.2 are required. The integral equations (4.11), (5.11), (6.11) and (7.10) can be generally written as Sj (t) +

m l=1

and

Jl∗

ηj (t) Ap,l (s) 1 Im Sl (s) ds = 2φj (t) π Ap,j (t) ηj (t) − ηl (s) m l=1

Jl∗


(8.6)

(8.7)

for t ∈ Jj∗ , where only the functions φj (t) and the constants h˜ j in the right-hand sides of (8.6) and (8.7) are different from one integral equation to another, j = 1, 2, . . . , m.

...................................................

δj

15


δj (t) : [0, 2π ] → [0, 2π ] and


By introducing t = δj (τ ) and s = δl (σ ), we have m ηj (δj (τ )) Ap,l (δl (σ )) 1 Im Sj (δj (τ )) + Sl (δl (σ ))δl (σ ) dσ Ap,j (δj (τ )) ηj (δj (τ )) − ηl (δl (σ )) Jl∗ π

16

and

(8.8) m Jl∗

l=1

Jj,l (δj (τ ), δl (σ ))Sl (δl (σ ))δl (σ ) dσ = h˜ j ,

(8.9)

for τ ∈ Jj∗ . Multiplying both sides of equation (8.8) by δj (τ ) for τ ∈ Jj∗ , we get m ηj (δj (τ ))δj (τ ) Ap,l (δl (σ )) 1 Im Sj (δj (τ ))δj (τ ) + Sl (δl (σ ))δl (σ ) dσ Ap,j (δj (τ )) ηj (δj (τ )) − ηl (δl (σ )) Jl∗ π l=1

= 2φj (δj (τ ))δj (τ ) and

(8.10)

m Jl∗

l=1

Jj,l (δj (τ ), δl (σ ))Sl (δl (σ ))δl (σ ) dσ = h˜ j .

(8.11)

Equations (8.10) and (8.11) are valid even when δj (τ ) = 0, i.e. for all τ ∈ Jj , j = 1, 2, . . . , m. By using (8.3) and (8.5), Sˆ (τ ) = S (δj (τ ))δ (τ ), φˆ j (τ ) = φj (δj (τ ))δ (τ ), Ap,j (δj (τ )) are constant and as δ : J → J is j

j

j

j

bijective and the function hj (t) are constant in each interval Jj , j = 1, . . . , m, we have m 2π ζj (τ ) Ap,l (σ ) 1 Im Sˆ j (τ ) + Sˆ l (σ ) dσ = 2φˆ j (τ ) Ap,j (τ ) ζj (τ ) − ζl (σ ) 0 π

(8.12)

l=1

and

m 2π l=1

0

Jj,l (τ , σ )Sˆ l (σ ) dσ = h˜ j .

(8.13)

Hence, we have Sˆ j (τ ) +

m 2π l=1

0

∗ [N2,j,l (τ , σ ) + Jj,l (τ , σ )]Sˆ l (σ ) dσ = 2φˆ j (τ ) + h˜ j (τ ),

τ ∈ Jj .

(8.14)

Integral equation (8.14) can be solved by means of Nyström method with trapezoidal rule. Integral equation (3.7) can be modified by using the same procedure as described above. Hence, the piecewise constant functions hk and νk can be computed from (3.9) and (3.10). In this paper, we choose test regions with connectivities three and four. The computations were carried out on Windows 7 64-bit operating system, Intel processor Quad-core 2.33GHz, 4GB DDR3 RAM using algorithms coded in Matlab R2011a. Example 8.1. Consider an unbounded region Ω − bounded by three circles

and where 0 ≤ t ≤ 2π .

Γ1 : η1 (t) = 2 + e−it , √ Γ2 : η2 (t) = −1 + i 3 + 0.5 e−it √ Γ3 : η3 (t) = −1 − i 3 + 1.5 e−it ,

√ For this example, the special points are z1 = 2, z2 = −1 + i 3 and α = 0. We choose the value of θ1 = π/2, θ2 = π/2 and θ3 = π/4. Figure 3 shows the images of the conformal mappings of the original region onto the canonical regions by using our method with n = 256 points. This example has also been considered in [15] for θ1 = π/2, θ2 = 0 and θ3 = 0 for the last image. The values of the parameters for the canonical regions for our method and [15] are given in table 1. Table 2

...................................................

= 2φj (δj (τ ))


l=1


(a)

(b)

(c)

17

5 4

1 0 0 –1

–5

5

–1.0

–0.5

–2 –3 –4

–0.2 –0.4 –0.6 –0.8 –1.0

0

0.5

1.0

–1.0

–0.5

–0.2 –0.4 –0.6 –0.8 –1.0

0

0.5

1.0

–5

(d)

(e) 4

5

3

4

2

3

1

2

0

–5

0

1

5

–2

0 0 –1

–3

–2

–4

–3

–5

–4

–6

–5

–1

–5

5

Figure 3. The original region Ω − and its images for example 8.1. (a) Ω − , (b) U1 , (c) U2 , (d) U3 and (e) U4 . (Online version in colour.)

Table 1. The values for approximated parameters in example 1 for spiral slit region with θ1 = π/2, θ2 = 0 and θ3 = 0. n

parameters

our method

Amano & Okano [15]

16

R1 R2

1.466503772756291 2.395214401312238

1.4683 2.393

R3

−2.226956363608526

−2.2259

..........................................................................................................................................................................................................

32

R1 R2

1.467544586417223 2.394496917138897

1.46757 2.39447

R3

−2.226349688304721

−2.22633

..........................................................................................................................................................................................................

64

R1

1.467540607962205

1.467540618

R2 R3

2.394498620150213 −2.226352161572016

2.39449861 −2.226352156

..........................................................................................................................................................................................................

128

R1

1.467540608209702

1.46754060820969

R2 R3

2.394498619975067 −2.226352161424804

2.394498619975067 −2.22635216142479

..........................................................................................................................................................................................................

shows the time taken for the computer to run the program coded in Matlab. The times taken are in seconds. The inverse transformations for each canonical region are given in figure 4a–d. The comparisons between the condition number of the matrices of our method and [15] are given in figure 5.

...................................................

2

1.0 0.8 0.6 0.4 0.2 0


3

1.0 0.8 0.6 0.4 0.2 0


(a)

1.0

–5

(b) 1.0 0.8 0.6 0.4 0.2 0 –1.0 –0.5–0.20 –0.40 –0.6 –0.8 –1.0

5

(c)

–5

0.5

18

5 4 3 2 1 0 –10 –2 –3 –4 –5

1.0 –5

5

(d) 4 3 2 1 0 0 –1 –2 –3 –4 –5 –6

5 –5

4 3 2 1 0 0 –1 –2 –3 –4 –5 –6

5 4 3 2 1 0 0 –1 –2 –3 –4 –5

5 –5

5 –5

5 4 3 2 1 0 0 –1 –2 –3 –4 –5

5

Figure 4. The inverse of conformal mapping. (a) The inverse of U1 , (b) the inverse of U2 , (c) the inverse of U3 and (d) the inverse of U4 . (Online version in colour.)

condition number

108

adj. G.N.K. 1 adj. G.N.K. 2 C.S

106

104

102

n

102

Figure 5. Condition numbers of the matrices for adjoint generalized Neumann kernel formed with A1 , A2 and the charge simulation method [15], for example 8.1. Table 2. Time taken in seconds for computing the conformal mapping onto its canonical regions for example 1. n

U1

U2 (s)

U3 (s)

U4 (s)

128

0.603617

0.605556

0.612396

0.589240

256

1.172543

1.173658

1.149271

1.122841

512

3.025301

3.020078

3.027725

3.003177

1024

10.571163

10.421955

10.610214

10.432853

.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................

Example 8.2. Consider an unbounded region Ω − bounded by four rectangles Γ1 : η1 (t) = {x + iy : |x| = 2, |y − 3| ≤ 1} ∪ {x + iy : |x| ≤ 2, |y − 3| = 1}, Γ2 : η2 (t) = {x + iy : |x| = 2, |y + 3| ≤ 1} ∪ {x + iy : |x| ≤ 2, |y + 3| = 1},

...................................................

0.5

5 4 3 2 1 0 0 –1 –2 –3 –4 –5


1.0 0.8 0.6 0.4 0.2 0 –1.0 –0.5–0.20 –0.40 –0.6 –0.8 –1.0


(b)

(a)

0.8

6

0.6

0.6

4

0.4

0.4

2

0.2

0.2

0

0

–2

0

5

10

–1.0

–0.5

–0.2

0

0.5

1.0

–1.0

–0.5

0 –0.1

–4

–0.4

–0.4

–6

–0.6

–0.6

–8

–0.8

–0.8

–1.0

–1.0

0

0.5

1.0

(e)

(d)

–10

10 8 6 4 2 0 0 –2 –4 –6 –8 –10

–5

8 6 4 2 5

10

–10

0

–5

–2

0

5

10

–4 –6 –8

Figure 6. The original region Ω − and its images for example 8.2. (a) Ω − , (b) U1 , (c) U2 , (d) U3 and (e) U4 . (Online version in colour.)

(a)

(b)

5 1.0

–0.5

0 –0.2

0

0.5

1.0

–0.4

5

–1.0

–0.5

1.0

0

0

–5.0

–2 –3 –4

–1.0

–5

(c)

–5

(d )

10 8

8

6

6

4

4

–5

0

6

6

4

4

2

2

2

2 –10

5.0

–1

–0.8

–4

–1.0

0.5

–0.6

–3

–0.8

0 0 –0.2 –0.4

–2

–0.6

1

0.2

0 0 –1

–5.0

2

0.4

1

0.2

3

0.6

2

0.4

4

0.8

3

0.6

–1.0

5 1.0

4

0.8

0

5

–2

10

–10

–5

0

5

10

–6

–4

–2

0

0

2

4

6

–6

–4

–2

0 0

2

4

6

2

–4

4

–6

–6

–8

–8

–10

0

–2

–2

–4

–4

–6

–6

Figure 7. The inverse of conformal mapping. (a) The inverse of U1 , (b) the inverse of U2 , (c) the inverse of U3 and (d) the inverse of U4 . (Online version in colour.) Γ3 : η3 (t) = {x + iy : |x + 2| = 1, |y| ≤ 1} ∪ {x + iy : |x + 2| ≤ 1, |y| = 1} and

Γ4 : η4 (t) = {x + iy : |x − 2| = 1, |y| ≤ 1} ∪ {x + iy : |x − 2| ≤ 1, |y| = 1},

where 0 ≤ t ≤ 2π . This example has been considered in [37]. For example 8.2, the special points are z1 = 2, z2 = −2 and α = 0 and we choose the value of θj by θ1 = π/2, θ2 = π/2, θ3 = 0 and θ4 = 0. Figure 6 shows the images of the conformal mappings of the original region onto its canonical regions by using our method with n = 256 points. The inverse transformations for each canonical region are given in figure 7. Table 3 shows the time taken for computing the conformal mapping onto its canonical region between our proposed method and by the study of Nasser [37].

...................................................

–5

19

1.0

0.8


–10

(c)

1.0

8


(a)

(b)

1.5

–0.5

0

0

0.5

1.0

1.5

2.0

–1.0

0.5 –1.0

0.5

–1.5 –1.0 –0.5

0

0.5

1.0

1.5

2.0

–1.0

0.5 –1.0

0.4 0.2 0 –0.5 0 –0.2 –0.4 –0.6

0.5

1.0

–0.8 –1.0

–1.5

(e)

1.0

1.0 0.8 0.6

1.0

–2.0

0.5

(d )

1.5

0

0.4 0.2 0 0 –0.2 –0.4 –0.6 –0.8 –1.0

–1.5

(c)

–0.5

...................................................

–1.5 –1.0


0.5

–2.0

20

1.0 0.8 0.6

1.0

1.5 1.0 0.5

–2.0 –1.5 –1.0

–0.5

0

0

0.5

1.0

1.5

2.0

0.5 –1.0 –1.5

Figure 8. Image transformation by the conformal mapping of unbounded multiply connected region. (a) The original region, (b) U1 with θ = (π/2, π/2, 0, π/3), (c) U3 with θ = (π/2, π/2, π/4, 0), (d) U2 with θ = (π/2, 0, π/2, π/3) and (e) U4 with θ = (π/2, π/2, π/4, 0). (Online version in colour.)

(a)

(b)

1.5 1.0 0.8

1.0

0.6 0.5

0.4 0.2 –1.0

–0.5

0 0 –0.2

0.5

–2.0

–1.5 –1.0

–0.5

0

0

0.5

1.0

1.5

2.0

–0.4 –0.6

–1.0

–0.8 –1.0

–1.5

Figure 9. Inverse transformation for U1 . (a) Grid lines and (b) images. (Online version in colour.)

Table 3. Time taken in seconds for computing the conformal mapping onto its canonical regions for example 2. n

methods

128

ours

U1 (s) 0.833566

U2 (s) 0.806499

U3 (s) 0.829227

U4 (s) 1.644461

........................................................................................................................................................................

Nasser [37]

3.545890

3.699722

3.614476

3.701283

ours

1.708097

1.712417

1.741601

3.409029

..........................................................................................................................................................................................................

256

........................................................................................................................................................................

Nasser [37]

14.82121

15.24135

14.67791

14.93859

ours

4.935630

4.957821

4.917135

8.558020

..........................................................................................................................................................................................................

512

........................................................................................................................................................................

Nasser [37]

73.11641

76.08164

73.28085

73.60038

ours

18.43652

18.76065

18.43312

29.35082

..........................................................................................................................................................................................................

1024

........................................................................................................................................................................

Nasser [37]

424.1434

432.2561

427.4205

420.0699

..........................................................................................................................................................................................................

–0.6 –0.8 –1.0

0.5

1.0

–1.5

–1.0

0 –2.0 –1.5 –1.0 –0.5 0 –0.5

0.5 0.5 1.0 1.5 2.0

0.5 0 –2.0 –1.5 –1.0 –0.5 0 –0.5 –1.0 –1.5

0.5

0 –2.0 –1.5 –1.0 –0.5 0 –0.5

–1.0

–1.5

0.5 1.0 1.5 2.0

0.5

0.5

–1.5

–1.5

0.5 1.0 1.5 2.0


–1.0

–1.0

0 0.5 1.0 1.5 2.0 –2.0 –1.5–1.0 –0.5 0 –0.5

1.0

1.0

0 –2.0 –1.5 –1.0 –0.5 0 –0.5

1.5

1.5


1.0

1.0

0.5 1.0 1.5 2.0

1.5

1.5


–1.0

0.2 0 –0.5 0 –0.2 –0.4

1.0

1.5

...................................................


1.0 0.8 0.6 0.4


21


Example 8.3. Consider an unbounded region Ω − bounded by

Γ2 : η2 (t) = {x + iy : |x + 0.75| = 0.75, |y − 0.625| ≤ 0.125} ∪ {x + iy : |x + 0.75| ≤ 0.75, |y − 0.625| = 0.125}, Γ3 : η3 (t) = 1.25 + 0.7i + 0.15 e−it and

Γ4 : η4 (t) = 1.4 − 0.05i + 0.2 e−it ,

where 0 ≤ t ≤ 2π . For this example, we show some simple image transformation based on the presented method. The image transformation is done by Matlab’s function imtransform where the conformal map and the inverse conformal map algorithms are used to obtain the desired transformation; see figure 8 for image transformation based on the conformal maps and figures 9–12 for image transformation based on the inverse conformal maps. For this example, the special points are z1 = −0.7i, z2 = −0.8 + 0.6i and α = 0.4 + 0.1i. As the image transformation is based on conformal map, some details (angles and magnitude between curves of the grid lines) are preserved although the shape of the images are different.

9. Conclusion This paper presented two integral equations with adjoint generalized Neumann kernels for solving RH problems for certain auxiliary functions related to the conformal maps to the canonical regions. The integral equations are discretized using Nyström’s method with the trapezoidal rule. For regions with corners, we use a quadratic formula based on a graded mesh. The resulting linear systems are solved by Gaussian elimination in Matlab. Several examples are given to show the effectiveness of the present method. The advantage of this method is that it allows to compute the values of the conformal mapping as well as the values of its inverse. Funding statement. This work was financially supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia (GUP Q.J130000.2526.04H62 and Q.J130000.2426.01G11).

References 1. Kerzman N, Stein E. 1978 The Cauchy kernel, the Szegö kernel, and the Riemann mapping function. Math. Ann. 236, 85–93. (doi:10.1007/BF01420257) 2. Kerzman N, Trummer M. 1986 Numerical conformal mapping via the Szegö kernel. J. Comput. Appl. Math. 14, 111–123. (doi:10.1016/0377-0427(86)90133-0) 3. Razali MRM, Nashed MZ, Murid AHM. 1997 Numerical conformal mapping via the Bergman kernel. J. Comput. Appl. Math. 82, 333–350. (doi:10.1016/S0377-0427(97)00091-5) 4. Murid AHM, Razali MRM. 1999 An integral equation method for conformal mapping of doubly-connected regions. Matematika 15, 79–93. 5. Murid AHM, Nashed MZ, Razali MRM. 1998 Numerical conformal mapping for exterior regions via the Kerzman–Stein kernel. J. Integral Equ. Appl. 10, 517–532. (doi:10.1216/ jiea/1181074250) 6. Razali MRM, Nashed MZ, Murid AHM. 2000 Numerical conformal mapping via the Bergman kernel using the generalized minimum residual method. Comput. Math. Appl. 40, 157–164. (doi:10.1016/S0898-1221(00)00149-8) 7. Sangawi AWK, Murid AHM, Nasser MMS. 2011 Linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits. Appl. Math. Comput. 218, 2055–2068.

...................................................

∪ {x + iy : |x| ≤ 0.725, |y + 0.725| = 0.125},


Γ1 : η1 (t) = {x + iy : |x| = 0.725, |y + 0.725| ≤ 0.125}

22


23 ...................................................


8. Sangawi AWK, Murid AHM, Nasser MMS. 2012 Annulus with circular slit map of bounded multiply connected regions via integral equation method. Bull. Malays. Math. Sci. Soc. 2 35, 945–969. 9. Sangawi AWK, Murid AHM, Nasser MMS. 2012 Circular slits map of bounded multiply connected regions. Abstr. Appl. Anal. 2012, 970928. (doi:10.1155/2012/970928) 10. Sangawi AWK, Murid AHM, Nasser MMS. 2012 Parallel slits map of bounded multiply connected regions. J. Math. Anal. Appl. 389, 1280–1290. (doi:10.1016/j.jmaa.2012.01.008) 11. Sangawi AWK, Murid AHM, Nasser MMS. 2013 Radial slits map of bounded multiply connected regions. J. Sci. Comput. 55, 309–326. (doi:10.1007/s10915-012-9634-3) 12. Yunus AAM, Murid AHM, Nasser MMS. 2012 Conformal mapping of unbounded multiply connected regions onto canonical slit regions. Abstr. Appl. Anal. 2012, 293765. (doi:10.1155/2012/293765) 13. Amano K, Okano D, Ogata H, Sugihara M. 1994 A charge simulation method for the numerical conformal mapping of interior, exterior and doubly connected domains. J. Comput. Appl. Math. 53, 353–370. (doi:10.1016/0377-0427(94)90063-9) 14. Amano K, Okano D, Ogata H, Sugihara M. 2003 Numerical conformal mappings of unbounded multiply-connected domains using the charge simulation method. Bull. Malays. Math. Sci. Soc. 2 26, 35–51. 15. Amano K, Okano D. 2010 A circular and radial slit mapping of unbounded multiply connected domains. JSIAM Lett. 2, 53–56. 16. DeLillo TK, Benchama N. 2003 A brief overview of Fornberg-like methods for conformal mapping of simply and multiply connected regions. Bull. Malays. Math. Sci. Soc. 2 26, 53–62. 17. Benchama N, DeLillo T, Hrycak T, Wang L. 2007 A simplified Fornberg-like method for the conformal mapping of multiply connected regions-comparisons and crowding. J. Comput. Appl. Math. 209, 1–21. (doi:10.1016/j.cam.2006.10.030) 18. Crowdy D, Marshall J. 2006 Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6, 59–76. (doi:10.1007/BF03321118) 19. Czapla R, Mityushev V, Rylko N. 2012 Conformal mapping of circular multiply connected domains onto slit domains. Electron. Trans. Numer. Anal. 39, 286–297. 20. DeLillo T, Horn M, Pfaltzgraff J. 1999 Numerical conformal mapping of multiply connected regions by Fornberg-like methods. Numer. Math. 83, 205–230. (doi:10.1007/s002110050447) 21. Fornberg B. 1980 A numerical method for conformal mappings. SIAM J. Sci. Stat. Comput. 1, 386–400. (doi:10.1137/0901027) 22. Henrici P. 1986 Applied and computational complex analysis, vol. 3. New York, NY: John Wiley. 23. Nasser MMS. 2009 A boundary integral equation for conformal mapping of bounded multiply connected regions. Comput. Methods Funct. Theory 9, 127–143. (doi:10.1007/BF03321718) 24. Nasser MMS. 2009 Numerical conformal mapping via boundary integral equation with the generalized Neumann kernel. SIAM J. Sci. Comput. 31, 1695–1715. (doi:10.1137/070711438) 25. Nasser MMS, Murid AHM, Sangawi AWK. 2013 Numerical conformal mapping via a boundary integral equation with the adjoint generalized Neumann kernel. (http://arxiv.org/abs/1308.3929). 26. Nasser MMS. 2013 Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions. J. Math. Anal. Appl. 398, 729–743. (doi:10.1016/j.jmaa.2012.09.020) 27. Papamichael N, Kokkinos C. 1984 The use of singular functions for the approximate conformal mapping of doubly-connected domains. SIAM J. Sci. Stat. Comput. 5, 684–700. (doi:10.1137/0905049) 28. Trefethen LN (ed.) 1986 Numerical conformal mapping. Amsterdam, The Netherlands: North Holland. 29. Wegmann R. 2001 Fast conformal mapping of multiply connected regions. J. Comput. Appl. Math. 130, 119–138. (doi:10.1016/S0377-0427(99)00387-8) 30. Wegmann R. 2005 Methods for numerical conformal mappings. In Handbook of complex analysis: geometric function theory, vol. 2 (ed. R Kühnau), pp. 351–477. Amsterdam, The Netherlands: Elsevier B. V. 31. Yunus AAM, Murid AHM, Nasser MMS. 2013 Numerical evaluation of conformal mapping and its inverse for unbounded multiply connected regions. Accepted for publication in: Bull. Malays. Math. Sci. Soc. 2012, 293765. 32. Crowdy D. 2012 Conformal slit maps in applied mathematics. Anziam J. 53, 171–189. (doi:10.1017/S1446181112000119)


24 ...................................................


33. Sakajo T. 2009 Equation of motion for point vortices in multiply connected circular domains. Proc. R. Soc. A 465, 2589–2611. (doi:10.1098/rspa.2009.0070) 34. Schinzinger R, Laura PAA. 1991 Conformal mapping: methods and applications. Amsterdam, The Netherlands: Elsevier. 35. Wang Y, Gu X, Chan T, Thompson P, Yau S. 2006 Brain surface conformal parameterization with algebraic functions. In MICCAI 2006, Part II, pp. 945–969. Lecture Notes in Computer Science. Springer. 36. Koebe P. 1916 Abhandlungen zur theorie der konfermen abbildung. iv. abbildung mehrfach zusammenhängender schlichter bereiche auf schlitzbereiche. Acta Math. 41, 305–344. (doi:10.1007/BF02422949) 37. Nasser MMS. 2011 Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonical slit domains. J. Math. Anal. Appl. 382, 47–56. (doi:10.1016/j.jmaa.2011.04.030) 38. Nehari Z. 1952 Conformal mapping. New York, NY: Dover. 39. Wen GC. 1992 Conformal mapping and boundary value problems (English transl. of Chinese edition 1984). Providence, RI: American Mathematical Society. 40. Nasser MMS, Al-Shihri FAA. 2013 A fast boundary integral equation method for conformal mapping of multiply connected regions. SIAM J. Sci. Comput. 35, A1736–A1760. (doi:10.1137/120901933) 41. DeLillo T, Driscoll T, Elcrat A, Pfaltzgraff J. 2008 Radial and circular slit maps of unbounded multiply connected circle domains. Proc. Math. Phys. Eng. Sci. 464, 1719–1737. (doi:10.1098/rspa.2008.0006) 42. Nasser MMS, Murid AHM, Zamzamir Z. 2008 A boundary integral method for the Riemann– Hilbert problem in domains with corners. Complex Var. Eliptic Equ. 53, 989–1008. (doi:10.1080/ 17476930802335080) 43. Nasser MMS, Murid AHM, Ismail M, Alejaily EMA. 2011 Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions. Appl. Math. Comput. 217, 4710–4727. (doi:10.1016/j.amc.2010.11.027) 44. Wegmann R, Nasser MMS. 2008 The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions. J. Comput. Appl. Math. 214, 36–57. (doi:10.1016/j.cam.2007.01.021) 45. Nasser MMS. 2009 The Riemann–Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions. Univ. Res. J. 20, 47–60. 46. Gakhov FD. 1966 Boundary value problem (English transl. of Russian edition 1963). Oxford, UK: Pergamon Press. 47. Atkinson KE. 1997 The numerical solution of integral equations of the second kind. Cambridge, UK: Cambridge University Press. 48. Kress R. 1990 A Nyström method for boundary integral equations in domains with corners. Numer. Math. 58, 145–161. (doi:10.1007/BF01385616)