CC BY
77Probl. Anal. Issues Anal. Vol. 8 (26), No 1, 2019, pp. 3-16 3
DOI: 10.15393/j3.art.2019.5050
UDC 517.54
D. F. Abzalilov, E. A. Shirokova
THE APPROXIMATE CONFORMAL MAPPING ONTO MULTIPLY CONNECTED DOMAINS
Abstract. The method of boundary curve reparametrization is generalized to the case of multiply connected domains. We construct the approximate analytical conformal mapping of the unit disk with N circular slits or an annulus with (N — 1) circular slits onto an arbitrary (N + 1) multiply connected finite domain with a smooth boundary. The method is based on the solution of the Fredholm equation. This solution is reduced to the solution of a linear system with unknown Fourier coefficients. The approximate mapping function has the form of a set of Laurent polynomials in the set of annular regions The method is easily computable.
Key words: conformal mapping, multiply connected domain, Fredholm integral equation
2010 Mathematical Subject Classification: 30C20, 30C30, 45B05
1. Introduction. Conformal mappings by analytical functions of complex variable play an important role in solving many problems of mechanics and mathematics, particulary in the case of plane potential fields and the Laplace equation [1]. The conformal mapping of circular domains (a disk with circular slits or an annulus with circular slits) onto a multiply connected domain with a complex boundary can be applied to solving plane boundary value problems for the corresponding domains with the help of Schottky-Klein prime functions known for circular domains [2]. Computer progress has stimulated appearance of many numerical methods for constructing conformal mapping [3]. However, all these methods are rather time-consuming. For example, the widely used Wegmann numerical method is based on solving the Riemann-Hilbert problem and involves iteration processes [4,5].
© Petrozavodsk State University, 2019
There are several types of canonical regions for conformal mappings [6]. The five types of canonical slit regions are a disk with concentric circular slits, an annulus with concentric circular slits, unbounded circular slit regions, unbounded radial slit regions, and unbounded parallel slit regions. Recenaly a few approaches to map bounded and unbounded multiply connected regions onto these five canonical regions by reformulating the mapping problems as Riemann-Hilbert problems have been proposed; these problems are solved by means of boundary integral equations with the generalized Neumann kernel [7-13]. The integral equations were dis-cretized by the Nystrom method with the trapezoidal rule used to obtain linear systems. The right-hand side of the integral equation in [7, 8] involves integral with the cotangent singularity which is approximated by Wittich's method.
Here we present a new method of the approximate conformal mapping of a unit disk with circular slits or an annulus with circular slits onto a multiply connected domain with a smooth boundary. We apply the integral equation with the Neumann kernel obtained from the necessary and sufficient condition for a function defined at the points of a smooth contour to be the boundary values of some function analytical in the correspondent domain. The theory of this method for simply connected domains has been developed in [14,15], the generalization to the case of a doubly connected domain and the relative numerical implementation can be found in [16]. Here we generalize the method to the case of multiply connected domains whose internal borders are images of circular arcs and an inner circle. The method is based on the boundary curves reparametrization. We do not search for corrections of boundary parametrization with linearization of this process [17,18]. The advantages of our method are the following: the method does not use any auxiliary constructions (triangulation, circle packing, domain decomposition) or conformal mappings (the zipper algorithm, the Schwarz-Christoffel mapping), it does not use the accessory solutions of boundary value problems (the conjugate function method, the Wegmann method), it does not use iterations. We do not apply the collocation method or the Nystrom's method for solution of the integral equation.
The solution of integral equations in our method is reduced to the solution of an infinite linear system. We obtain an approximate mapping by solving the finite system with a truncated matrix. The smooth solution of the problem has the form of a Laurent polynomial in each annular
region. This form allows us to apply the differentiation techniques for solutions of some physical problems in the corresponding domains. We must note that our method is applicable only for the domains with smooth boundary components or the boundary components with vertices of the angles more than n. The cases of the angles less than n bring to kernels with strong singularities.
2. Approximate conformal mapping of the circular domains of two types onto a multiply connected domain by means of the boundary reparametrization. Consider a finite multiply connected domain Dz bounded by the outer simple smooth curve L0 and the inner simple smooth curves Ls given by the equations
Ls = {z = zs(t), zs(0) = zs(2n), t e [0, 2n]},s = 0,N.
Also, assume that complex representations of the boundary curves Ls are as follows:
Us
zs(t)= ^ dkseikt, t e [0,2n], s = 0,N.
k=-ms
The parametrization traces the domain Dz along L0 counterclockwise and along the inner contours Ls, s = 1,N, clockwise.
Definition 1. We call the unit disk |Z | < 1 with N circular slits Z = Rje%e, 9ij < 9 < 92j, with a constant Rj < 1, j = 1,N, an (N + 1)-connected circular domain of the first type.
Definition 2. We call the annulus r < |Z| < 1, with (N — 1) circular slits Z = Rjeie, 9ij < 9 < 92j, with a constant Rj e (r, 1), j = 1, N — 1, an (N + 1)-connected circular domain of the second type.
Existence theorems for the approximate conformal mappings of given (N + 1) multiply connected finite domain with a smooth boundary onto a circular domain of type I and of type II are well-known (see, e.g., [6,13,19]).
Theorem 1. There exists an (N + 1)-connected circular domain Dz of the first type and there exists a regular function z = f (() in this domain, such that the function f (Z) maps conformally the domain Dz onto the given (N + 1)-connected domain Dz with smooth boundary components. The map is unique under the following conditions: f (0) = Z0, f (1) = Zi, Z0 e Dz, Zi e L0. The corresponding approximate function can be constructed in the form of a Laurent polynomial.
Proof. We assume that 0 g Dz and Z0 = 0, without loss of generality. The proof is constructive. We construct the conformal map of the circular domain of the first type onto the domain Dz by reparametrization of the given boundary representations. So, we search for the function t0(B), B G [0, 2n], and for ts(B), s = I,N, B G [Bis, 02s], such that the values zs(ts(B)), s = 0, N, are the boundary values of an analytic function in the corresponding circular domain. The parameters Rs, 01s, 02s, are also unknown and will be found within the solution process.
Let us consider the analytic in the domain Dz function Z (z) that maps conformally the domain Dz onto D^ with the correspondence Z(0) = 0 and
„«ch^«D Acco^g to the „y and
z
sufficient condition for log ^ to be analytic in Dz is the boundary relation
N 2n
lo^-^sW = V IfloJ dT, (!)
Rseies(t) Z^ ni J ta \Raeie°(t) J z*(t) - Zs(t)
where t g [0, 2n] , s = 0, N, R0 = 1. _
We introduce the functions qs(t) = argzs(t) — 0s(t), s = 0, N, where 0s(t) is the polar angle of the image of the point of zs(t) and separate the imaginary part of both sides of equation (1):
N 2n
qs(t) = è 1 / (t) (arg[z*(t) — zs(t)])T dT— *=0 o
N 2n
— Ë 1 /log ^ (log (t) — Zs(t)|)T dT, s = 0,N.
*=° o
After differentiating this relation and integrating the result by parts, we obtain the following relations on the functions qS (t):
N 2n
qs (t) = è 1 f q* (T)K,s(T,t)dT + Ps(t), s = 0, N,
*=0 0
K*s(T,t) = — (arg[z*(t) — Zs(t)])t,
N 2n
Ps(t) = è 1 i[log |z*(T)|]' (log |z*(t) — Zs(t)|)t dT.
7r
*=0 0
The kernel (log |zCT(t) —zs(t)|)t has a singularity in the form of cot for a = s:
T-t
ns
(log |zs(t) - Zs(t)|)t = Re (log E dks[eikT - eikt])
k=—ms
k-1
1 T — f / I s
-4cot ^ +(log|£ dkseikt^ eil(T-t)-
k=1 l=0
2
E d
k=1
(-k)se
—ikT
k-1
E
l=0
il(T-t)
The Cauchy principal value integral
2n
If T — t
n J [log (t)|]' cot —dT
0
can be calculated via the Hilbert formula [20] as in [16].
We search for the approximate solution of (2) in the form of the Fourier polynomial
q's (t)
M
/E j ' j=i
ajs cos jt + /3js sin jt, t E [0, 2n].
(3)
Existence of exact solution to equation (2) and convergence of the approximate solution to the exact one as M ^ were proved in [15] for simply connected domains. This proof can be applied to the case of multiply connected domains if we replace the corresponding space l2 by the space l2 x l2 x ... x l2.
Now the integral Fredholm equations of the second kind (2) can be reduced to the linear system for the Fourier coefficients ajs and
( A00 B00 A01 B01
Coo D00 C01 D01
A10 B10 A11 B11
C10 D 10 C11 D 11
\CN0 DN 0 CN1 DN1 ■ ■ ■ DNN J \$N J \pN J
B0N \
D0N
B1N
D1N
(
$0 a1 $1
/ «0\
b0 a1 b1
t
where as = (aos,...,«Ms)T, Bs = (B0s,..., Bms)t. The vectors as = (a0s,..., aMs)T, bs = (b0s,..., bMs)T on the right-hand side of the system consist of the elements
2n 2n
I f _ , , I
ajs = — / ps(t) cos jtdt, bjs = — / ps(t) sin jtdt, j = I, M, s = 0, N. n J n J
oo
The block matrices Aas, Bas, CCTs, Das of size MxM consist of the elements
2n 2n
Aasjfc = ias^jfc - J cos krdr J KCTs(r,t) cos jtdt, a = 0, N, oo
2n 2n
BCTsjk =--2 y sin krdr J KCTs(r,t)cos jtdt, s = 0, N,
oo
2n 2n
I
CCTsjk =--cos krdr J KCTs(r,t)sin jtdt, j = I,M,
oo
2n 2n
D^sjfc = ¿as^fc - y sin krdr j KCTs(r,t)sin jtdt, k =I,M, oo
where and j are the Kronecker delta functions.
The functions qs(t), s = 0, N, can be restored via their derivatives (3) with an arbitrary constant summand
m B
qs(t) = qos + <?s(t), qs(t) = V j sin jt - Bjs cos jt, t g [0, 2n]. (4)
We choose the constant summand q00 in accordance with the condition f (I) = Z1 in the following way. We find the value of the parameter i such
that Z0(i) = Zi. Now <200 = arg(Zi) - q0( i). _
We obtain the values of the other constant summands q0s, s = I,N, and also the values of Rs, s = I,N, in the following way. We take N points in each of the N finite components of the set complement of Dz. Let us denote these points z*, j = I,N. The function log(zs(t)/Rsei0s(i)), s = 0, N, is the boundary value of the analytical in Dz function, so the Cauchy integral with the corresponding density along the boundary of Dz
vanishes at the points z*, j = 1, N. Therefore, we have the linear complex system
N 2n
E / (^Os - log + log |zs(r)| + iqs(T)) [log(zCT(r) - z*)]Tdr = 0,
a=°0
j = 1, N, with the unknown real q0s and log Rs, s = 1, N.
We restore the values of 91j and 92j, j = 1, N, after we have restored q0j. Indeed,
dij = min [arg zi(t) - qj(t)] - qoj, $2j = max [arg Zi(t) - qj(t)] - qoj.
t€[0,2n] iG[0,2n]
So, all parameters of the circular domain of the first type Dz are found.
Now we have the functions qs(t), t g [0, 2n], s = 0, N, and, therefore, we can restore the relations between the boundary parameters of the domains Dz and Dz via the formula 9s(t) = arg zs(t) - qs(t). Note that 90(t) grows monotonically when t grows from 0 to 2n, 90(2n) -00(O) = 2n, while each of the functions ds(t), s = 1,N, is 2n-periodic with one interval of increase and one interval of decrease. We can restore the inverse to 90(t), monotonically increasing the function t0(9), and we can restore the single-valued functions t±(9), 9 g [91s,92s], where t± denotes the parameter of a twice traversed circular slit.
The approximate analytical function that maps Dz onto Dz now has the form of the Cauchy integral
2n Ar 02s
1 f Z0(t0(9))ei0d9 + ^ 1 y[zs(t+(9)) - Zs(t-(9))]Rsei0
f(Z) = eie - z Rses-Z d9.
0 s=1 01s
We can apply the Cauchy integral in the form
2n Ar 2n
f(z) = ±_ f z0(t)ei0o(t)9> (t)dt + ^ 1_ [ zs(t)Rsei0s(t)9> (t)
f (Z ) 2nJ ei0°(t) - Z Rsei0s(t) - Z d
0 s=1 0
in order not to deal with the functions t± (9) and not to integrate along the different borders of the same slit.
It is worthy to apply the Laurent series expansions instead of the Cauchy integral for the representation of the function z(Z). We have, for
Z g Dz, |Z| = R., s = 0, N, the series representation of the integrals in (5):
at 2n
N ro
N ro Z k r
f (Z)= E E ¿W z.(t)e—ik*s(i)0.(t)dfc-
s=0 fc=0 s ^
|Z|<Rs 0
s=0 fc=0 " s
N ro -ofc
2n
£ f (t)dt.
s=1 fc=0 0
|Z|>Rs 0
We have, for Z g Dc, Z = R, j g {1, 2,..., N}, 0 g [01,, 02,], the following representation
i 1 f z, (t)ei(ij(t)—*)/20j(t)dt| f (Rje )~
4niJ sin((0,(t) - 0)/2) 0
2n
N ro Rfc gifc0
+ z.(t)e-ik»-(')0:,(t)dt-
2nRk
s=0 fc=0 s
0
2n
E E ^Rr z^e^V.(t)dt.
2nRk „
s=1 fc=0 j 0
Rj >RS
Theorem 1 is proved. □
Theorem 2. There exists an (N+ 1)-connected circular domain Dz of the second type and there exists a regular function z = f (Z) in this domain such that the function f (Z) maps conformally the domain Dz onto the given (N + 1)-connected domain Dz with smooth boundary components. The map is unique under the following conditions: the image of the inner circle |Z| = r is the boundary component L,, j g {1, 2,... ,N}, f (1) = Z1, Z1 g L0. The corresponding approximate function can be constructed in the form of a Laurent polynomial.
2n
Proof. We assume that j = N and J(arg zN(t))'dt = —2n without loss of
0
generality. We construct the conformal map of the circular domain of the first type onto the domain Dz by reparametrization of the given boundary
Rj <RS
representations. So, we search for functions ts(9), 9 E [0, 2n], s = 0, N, and for functions ts(9), 9 E [91s, 92s], s = 1, N — 1. The construction is as that for mapping of the circular domain of the first type.
We consider the analytic in the domain Dz function Z(z) that maps conformally the domain Dz onto Dz, and the analytic in Dz function log |. We apply the boundary relation (1) as this has been done above. We introduce the functions qs(t) = argzs(t) — 9s(t), s = 0,N. After separation of the imaginary parts of the both sides of the previous equation, differentiation, and integration by parts, we get equation (2). We reduce the solution of the integral equation to the solution of a linear system with truncated matrices if we consider q's (t) representation (3). Now we have representation (4) for the functions qs(t). The constant summand q00 can be restored in the same way as for the previous case. The values of q0s and Rs, s = 1, N, can be also restored as has been done above with the help of the additional points z*, s = 1,N, located in the exterior of the domain Dz. Note that z*N = 0.
Finally, we have the mapping function in the series form
N—1 œ
f (Z)= E E ^¿RkJ z-Ve-*0'1^m-
s=0 k=0 s
\C\<R'
N œ -ok
2n
EE^I/ Zs(t)eikds^tt 9's (t)dt,
2nZ k
s=1 k=0
\Z\>R
s
0
for Z E Dc, |Z| = Rs, s = 0, N. □
We offer a new method of finding the values of q0s, Rs, s = 1, N, in order to avoid using many additional points z*, j = 1, N — 1, and to deal only with the point z*N = 0. Let 9s(t) = argzs(t) — qs(t). Now we have the following integral representation of the mapping function for Z E Dz:
2n Ar 2n -
1 f z0(t)e^9> (t)dt + ^ 1 f zs(t)Qseids(t)9's(t)
f (Z ) = 2ni ei6o{t) — Z + 2nJ Qse-s(t) — Z dt, (6) 0 s=1 0
where Qs = Rse iq0s, s = 1, N are unknown. The Cauchy integral vanishes at the points outside Dz. So, the expression at the right-hand side of (6)
and all derivatives of this expression vanish at the point Z = 0. Therefore, we have for all j g N
J (t)dt + ^ J ^(Q^^1-'^(t)dt = 0. (7)
0 s=1 0
We apply relations (7) in order to find the values of = Rse-iq0s, s = 1, N, and, therefore, the values of q0s, Rs, s = 1, N. The least-squares method can be applied. After we find these values, we apply formula (6) as the representation of the mapping function.
4. Examples of conformal mappings. In the first example, we apply the conformal mapping from [10]; the parametrization of the boundaries is as follows:
Lo ={z = 4 + 6i + (10 + 3 cos 3t)eit}, Li ={z = 0.5 + 12i + 0.5e-in/4(eit + 4e-it)}, L2 ={z = 9 + 6i + 0.5ein/4(ei4 + 4e-it)}, L3 ={z = 0.5 + 0.5ein/4(ei4 + 4e-it)}.
Recall that [10] describes construction of the conformal mapping of a bounded multiply connected region onto an annulus with circular slits and we are solving the inverse problem: we construct an approximate analytical conformal mapping of an annulus with circular slits onto the given multiply connected domain.
The result of the conformal mapping is shown in Figure 1.
Figure 1: Domains D and Dz in example 1.
We also investigated the effect of the number M of the Fourier coefficients on the parameters of the problem and accuracy.
Table 1: Accuracy and the elapsed time for example 1.
M Ri R2 R3 £ Time (s)
25 0.939091 0.888035 0.315839 0.4916 10—3 0.28
50 0.939829 0.889457 0.315620 0.3010 10—4 0.42
100 0.940017 0.889814 0.315564 0.1862 10—5 1.07
200 0.940063 0.889902 0.315550 0.1158 10—6 4.07
400 0.940074 0.889924 0.315546 0.7219 10—8 19.84
Table 1 presents the results; it also shows the elapsed time. Calculations were carried out on a computer Intel Core i5-3330 3Ghz CPU, 4Gb RAM, all the computations were done using Fortran (double precision).
Figure 2: Domains Dz and Dz in example 2.
Figure 2 shows an example of a conformal mapping on the domain with non-convex and non-starlike boundary. The boundary parametrization is
Lo ={z = eit — 0.5ie2it + 0.2e-2it},
L1 ={z = —0.55 — 0.4i — 0.1(1 — i)eit — 0.1(1 — 2i)e-it},
L2 ={z = —0.5 + i + 0.2e-it},
L3 ={z = 0.7i + (1 + i)(0.04eit + 0.1e-it) — 0.02e2it},
L4 ={z = 0.2 + 0.15i — 0.1(1 + i)eit — 0.3(1 — i)e-it — 0.03(1 — i)e2it}.
In the following example, we consider the case of 0 e Dz. The domain
is the disk with four cut out smaller disks: Lo = {z = 4eit:}, Li = {z = 2 + e-it}, L2 = {z = -2 + e-it}, L3 = {z = 2i + e-it}, L4 = {z = -2i + e-it}. An example of a conformai mapping onto the domain is shown by Figure 3.
Figure 3: Domains D^ and Dz in example 3.
5. Conclusion. It is possible to construct the approximate conformal mapping of the unit disk or an annulus with circular slits onto the given multiply connected domain by Laurent polynomial functions in each annular region. The solution can be reduced to a finite linear system. The method is easily computable.
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Received July 30, 2018.
In revised form, December 19, 2018.
Accepted December 20, 2018.
Published online January 5, 2019.
Kazan Federal University
18 Kremlyovskaya str Kazan, 420008, Russia
E-mail:
D. F. Abzalilov
E. A. Shirokova
