We consider a nonlinear integral equation which can be interpreted as a generalization of Theodorsen’s nonlinear integral equation. This equation arises in computing the conformal mapping between simply connected regions. We present a numerical method for solving the integral equation and prove the uniform convergence of the numerical solution to the exact solution. Numerical results are given for illustration.
1. Introduction
Numerical methods for conformal mapping from a simply connected region onto another simply connected region are available only when one of the region is a standard region, mostly the unit disk D. Let G and Ω be bounded simply connected regions in the z-plane and w-plane, respectively, such that their boundaries Γ:=∂G and L:=∂Ω are smooth Jordan curves. Then the mapping Ψ:G→Ω is calculated as the composition of the maps G→D→Ω.
Recently, a numerical method has been proposed in [1] for direct approximation of the mapping Ψ:G→Ω. Assume that Γ and L are star-like with respect to the origin and defined by polar coordinates
(1)η(t)=ρ(t)eit,ζ(t)=R(t)eit,0≤t≤2π,
respectively, such that both ρ and R are 2π-periodic continuously differentiable positive real functions with nonvanishing derivatives. By the Riemann-mapping theorem, there exists a unique conformal mapping function Ψ:G→Ω normalized by Ψ(0)=0, Ψ′(0)>0. The boundary value of the function Ψ is on the boundary L and can be described as
(2)Ψ+(η(t))=ζ(S(t)),0≤t≤2π,
where S(t) is the boundary correspondence function of the mapping function Ψ. The function S(t) is a strictly increasing function so that S(t)-t is a 2π-periodic function.
The function S(t) is the unique solution of a nonlinear integral equation which can be interpreted as a generalization of Theodorsen’s nonlinear integral equation [1]. The proof of the existence and the uniqueness of the solution of the nonlinear integral equation was given in [1] for regions Ω of which boundaries L=∂Ω satisfy the so-called ϵ-condition; that is,
(3)ε:=max0≤t≤2π|R′(t)R(t)|<1.
In this paper, the nonlinear integral equation is solved by an iterative method. Each iteration of the iterative method requires solving an n×n linear system which is obtained by discretizing the integrals in the integral equation by the trapezoidal rule. The linear system is solved by a combination of the generalized minimal residual (GMRES) method and the fast multipole method (FMM) in O(nlnn) operations. The main objective of this paper is to prove the uniform convergence of the numerical solution to the exact solution. We also study the properties of the generalized conjugation operator. Numerical results are presented for illustration.
2. Auxiliary Materials2.1. The Functions θ and τ
Let w=f(z) be the mapping function from the simply connected region G onto the unit disk D with the normalization f(0)=0 and f′(0)>0. Then the boundary value of the function f is on the unit circle and can be described as
(4)f(η(t))=eiθ(t),0≤t≤2π.
The function θ(t) is the boundary correspondence function of the mapping function f where θ(t)-t is a 2π-periodic function and θ′(t)>0 for all t∈[0,2π]. Let τ(t) be the inverse of the function θ(t). Then τ(t) is the boundary correspondence function of the inverse mapping function z=f-1(w) from D onto G; that is,
(5)f-1(eit)=η(τ(t)),0≤t≤2π,
where τ(t)-t is a 2π-periodic function and τ′(t)>0 for all t∈[0,2π].
2.2. The Norms
Let H be the space of all real Hölder continuous 2π-periodic functions on [0,2π]. With the inner product
(6)(γ,ψ)=12π∫02πγ(s)ψ(s)ds,
the space H is a pre-Hilbert space. We define the norm ∥·∥2 by
(7)∥γ∥2:=(γ,γ)1/2.
Since s=τ(t)and if t=θ(s), we have
(8)∥(θ′)1/2γ∥2=∫02πθ′(s)γ(s)2ds=∫02πγ(τ(t))2dt=∥γ∘τ∥2.
With the norm ∥·∥2, we define a norm ∥·∥θ by
(9)∥γ∥θ:=∥(θ′)1/2γ∥2=∥γ∘τ∥2.
We define also the maximum norm ∥·∥θ by
(10)∥γ∥∞:=max0≤t≤2π|γ(t)|.
Since θ(2π)-θ(0)=2π, we have
(11)∥γ∥θ2=12π∫02πθ′(t)γ2(t)dt≤∥γ∥∞212π∫02πθ′(t)dt=∥γ∥∞2,
which implies that
(12)∥γ∥θ≤∥γ∥∞.
Theorem 1.
If ∫02πθ′(s)γ(s)ds=0, then
(13)∥γ∥∞2≤2π∥γ∥θ∥γ′θ′∥θ.
Proof.
Since s=τ(t) and if t=θ(s), we have
(14)∫02π(γ∘τ)(t)dt=∫02πγ(τ(t))dt=∫02πθ′(s)γ(s)ds=0.
Thus, it follows from [2, page 68] that
(15)∥γ∘τ∥∞2≤2π∥γ∘τ∥2∥(γ∘τ)′∥2.
We have also
(16)∥(γ∘τ)′∥22=12π∫02π|(γ∘τ)′(t)|2dt=12π∫02πγ′(τ(t))2τ′(t)2dt=12π∫02πγ′(s)21θ′(s)ds=12π∫02πθ′(s)(γ′(s)θ′(s))2ds.
Hence,
(17)∥(γ∘τ)′∥22=∥γ′θ′∥θ2.
Since τ(·):[0,2π]→[0,2π] is bijective, we have
(18)∥γ∘τ∥∞≔max0≤t≤2π|γ(τ(t))|=max0≤τ≤2π|γ(τ)|=∥γ∥∞.
Hence, (15) and (18) imply that
(19)∥γ∥∞≤2π∥γ∘τ∥2∥(γ∘τ)′∥2.
Then (13) follows from (9), (17), and (19).
2.3. The Operators K and J
The conjugation operator K is defined by
(20)Kμ=∫02π12πcots-t2μ(t)dt.
Let J be the operator defined by
(21)Jμ=12π∫02πμ(t)dt.
Hence, the operators K and J satisfy [3]
(22)JK=0,K2=-I+J.
3. The Generalized Conjugation Operator
Let A be the complex 2π-periodic continuously differentiable function:
(23)A(s):=η(s).
We define the real kernels M and N as real and imaginary parts:
(24)M(s,t)+iN(s,t):=1πA(s)A(t)η′(t)η(t)-η(s).
The kernel N(s,t) is called the generalized Neumann kernel formed with A and η. The kernel N(s,t) is continuous and the kernel M has the representation
(25)M(s,t)=-12πcots-t2+M1(s,t),
with a continuous kernel M1. See [4] for more details.
We define the Fredholm integral operators N and M1 and the singular integral operator M on H by
(26)Nμ=∫02πN(s,t)μ(t)dt,M1μ=∫02πM1(s,t)μ(t)dt,Mμ=∫02πM(s,t)μ(t)dt.
We define an operator E on H by
(27)E=-(I-N)-1M.
The operator E is singular but bounded on H [1]. Finally, we define an operator Jθ by
(28)Jθμ=12π∫02πθ′(t)μ(t)dt.
Remark 2.
When Γ reduces to the unit, then θ′(t)=1, the operator Jθ reduces to the operator J, and the operator E reduces to the operator K; that is, the operator E is a generalization of the well-known conjugation operator K (see [1] for more details).
The operator E is related to the operator K by [1]
(29)μ=Eγiffμ∘τ=K(γ∘τ).
Since μ=(μ∘θ)∘τ and γ=(γ∘θ)∘τ, it follows from (29) that
(30)μ=Kγiffμ∘θ=E(γ∘θ).
Lemma 3 (see [1]).
Let γ,μ∈H be given functions. Then f(η(t))=γ(t)+iμ(t) is the boundary value of an analytic function in G with Imf(0)=0 if and only if
(31)μ=Eγ.
Lemma 4 (see [1]).
If γ∈H and μ=Eγ, then γ=c-Eμ with a real constant c=f(0) where f is the unique analytic function in G with the boundary values f(η(t))=γ(t)+iμ(t) and Imf(0)=0.
Lemma 5 (see [1]).
The operator E has the following properties:
(32)Null(E)=span{1},E3=-E,σ(E)={0,±i}.
Lemma 6.
The operator E has the norm
(33)∥E∥θ=1.
Proof.
The operator E has the norm ∥E∥θ≤1 [1]. Since i∈σ(E), hence 1=|i|≤∥E∥θ≤1. Hence, we obtain (33).
Lemma 7.
The operators Jθ and E satisfy
(34)JθE=0.
Proof.
For any γ∈H, let μ=Eγ, μ^=μ∘τ, and γ^=γ∘τ. Then, it follows from (29) that μ^=Kγ^. Thus,
(35)JθEγ(s)=Jθμ(s)=12π∫02πθ′(t)μ(t)dt=12π∫02πμ(τ(s))ds=12π∫02πμ^(s)ds=Jμ^(s)=JKγ^(s),
which by (22) implies that
(36)JθEγ(s)=0.
Since (36) holds for all functions γ∈H, the operator identity (34) follows.
Lemma 8.
The operator E satisfies
(37)E2=-I+Jθ.
Proof.
Let γ∈H and μ=Eγ. Then, by Lemma 4, γ=c-Eμ with a real constant c. By the definition of the operator Jθ, we have Jθc=c. Since JθE=0, we have
(38)Jθγ=Jθc-JθE=c.
Hence,
(39)E2γ=Eμ=-γ+c=(-I+Jθ)γ
holds for all γ∈H. Thus, the operator identities (37) follow.
Lemma 9.
For all functions γ∈H, we have
(40)∥Eγ∥θ≤∥γ∥θ,
with equality for all γ with Jθγ=0.
Proof.
For all functions γ∈H, the inequality (40) follows from (33).
For all functions γ with Jθγ=0, we have from (37) that γ=-E2γ. Hence
(41)∥γ∥θ=∥E2γ∥θ≤∥Eγ∥θ≤∥γ∥θ,
which means that ∥Eγ∥θ=∥γ∥θ.
Theorem 10.
Let γ∈H and μ=Eγ. Then
(42)μ′θ′=E(γ′θ′).
Proof.
For γ∈H and μ=Eγ, we have from (29) that μ∘τ=K(γ∘τ). Then, it follows from [2, page 64] that
(43)(μ∘τ)′=K(γ∘τ)′.
Hence, by (30), we have
(44)(μ∘τ)′∘θ=E((γ∘τ)′∘θ),
which implies that
(45)((μ∘τ)∘θ)′θ′=E(((γ∘τ)∘θ)′θ′).
Hence, we obtain (42).
4. The Generalized Theodorsen Nonlinear Integral Equation
The boundary correspondence function S(t) is the unique solution of the nonlinear integral equation
(46)S(t)-t=ElnR(S(·))ρ(·)(t)
which is a generalization of the well-known Theodorsen integral equation [1]. Nonlinear integral equation (46) can be solved by the iterative method
(47)Sk(t)-t=ElnR(Sk-1(·))ρ(·)(t),k=1,2,3,….
Then we have [1]
(48)∥Sk-S∥θ≤εk∥S0-S∥θ.
Thus, if the curve L satisfies the ε-condition (3), then
(49)∥Sk-S∥θ⟶0.
That is, the approximate solutions Sk(t) converge to S(t) with respect to the norm ∥·∥θ if ε<1.
In this section, we will prove the uniform convergence of the approximate solutions Sk(t) to the exact solution S(t). We will use the approach used in the proof of Proposition 1.5 in [2, page 69] related to Theodorsen’s integral equation. See also [5, 6].
Lemma 11.
Consider
(50)E[lnρ(·)](t)=t-θ(t).
Proof.
The function θ is the boundary correspondence function of the conformal mapping f from G onto the unit disk. Hence, the function θ(t)-t satisfies [1]
(51)θ(t)-t=Eln1ρ(·)(t).
Then (50) follows from (51).
The previous lemma implies that (46) can be rewritten as
(52)S(t)-θ(t)=E[lnR(S(·))](t),
and (47) can be rewritten as
(53)Sk(t)-θ(t)=E[lnR(Sk-1(·))](t).
Thus
(54)Sk(t)-S(t)=E[lnR(Sk-1(·))-lnR(S(·))](t).
Lemma 12.
Consider
(55)∥Sk-S∥θ≤(ε+∥S0-θ∥θ)εk.
Proof.
Let a be such that
(56)11+ε≤R(t)a≤1+ε,∀t.
Then
(57)|lnR(t)a|≤ln(1+ε)<ε,∀t.
Hence,
(58)∥lnRa∥∞<ε.
Thus
(59)∥E[ln(R(S(·))a)]∥θ≤∥E∥θ∥ln(R(S)a)]∥θ≤∥ln(R(S)a)]∥∞<ε.
Since
(60)S(t)-S0(t)=S(t)-θ(t)+θ(t)-S0(t)=E[lnR(S(·))](t)+θ(t)-S0(t)
and E(lna)=0, we have
(61)S(t)-S0(t)=E[ln(R(S(·))a)](t)+θ(t)-S0(t),
which implies that
(62)∥S-S0∥θ≤∥E[ln(R(S(·))a)]∥θ+∥θ-S0∥θ≤ε+∥θ-S0∥θ.
Hence (55) follows from (48).
Lemma 13.
Consider
(63)∥Sk′-S′θ′∥θ≤ε1-ε2(1+∥S0′θ′∥θ).
Proof.
We have
(64)∥Sk′-S′θ′∥θ=∥Sk′-θ′+θ′-S′θ′∥θ≤∥Sk′θ′-1∥θ+∥S′θ′-1∥θ.
Since
(65)∥S′θ′-1∥θ2=12π∫02πθ′(t)(S′θ′-1)2dt=12π∫02πθ′(t)(S′θ′)2dt-212π∫02πS′(t)dt+12π∫02πθ′(t)dt,∫02πS′(t)dt=2π, and ∫02πθ′(t)dt=2π, we obtain
(66)∥S′θ′-1∥θ2=∥S′θ′∥θ2-1.
Similarly, we have
(67)∥Sk′θ′-1∥θ2=∥Sk′θ′∥θ2-1.
In view of Theorem 10, it follows from (52) and (53) that
(68)S′(t)θ′(t)-1=E[R′(S(·))R(S(·))S′(·)θ′(·)](t),(69)Sk′(t)θ′(t)-1=E[R′(Sk-1(·))R(Sk-1(·))Sk-1′(·)θ′(·)](t).
Hence, it follows from (68) that
(70)∥S′θ′-1∥θ=∥E[R′(S(·))R(S(·))S′(·)θ′(·)]∥θ≤∥E∥θ∥R′(S(·))R(S(·))S′(·)θ′(·)∥θ≤ε∥S′θ′∥θ.
By (70) and (66), we have
(71)∥S′θ′-1∥θ2≤ε2∥S′θ′∥θ2=ε2+ε2∥S′θ′-1∥θ2.
Hence,
(72)∥S′θ′-1∥θ2≤ε21-ε2.
Similarly, it follows from (69) that
(73)∥Sk′θ′-1∥θ≤ε∥Sk-1′θ′∥θ,
which by (67) implies that
(74)∥Sk′θ′-1∥θ2≤ε2∥Sk-1′θ′∥θ2=ε2+ε2∥Sk-1′θ′-1∥θ2.
Hence,
(75)∥Sk′θ′-1∥θ2≤ε2+ε2∥Sk-1′θ′-1∥θ2≤⋯≤ε2+ε4+⋯+ε2k∥S0′θ′-1∥θ2,
which, in view of (67), implies that
(76)∥Sk′θ′-1∥θ2≤ε21-ε2(1+∥S0′θ′-1∥θ2)=ε21-ε2∥S0′θ′∥θ2.
Then (63) follows from (64), (72), and (76).
Theorem 14.
If ε<1, then the approximate solution Sk converges uniformly to the exact solution S with
(77)∥Sk-S∥∞≤2π1-ε2×(ε+∥S0-θ∥θ)(1+∥S0′θ′∥θ)εk/2+1/2.
Proof.
In view of (54), Lemma 7 implies that
(78)∫02πθ′(t)(Sk(t)-S(t))dt=0.
Thus, we have from (13) that
(79)∥Sk-S∥∞2≤2π∥Sk-S∥θ∥Sk′-S′θ′∥θ.
Hence (77) follows from (55) and (63).
The following corollary follows from the previous theorem.
Corollary 15.
If
(80)S0(t)=θ(t)=t-E[lnρ(·)](t),
then
(81)∥Sk-S∥∞≤2π1-ε2εk/2+1.
Remark 16.
When Γ reduces to the unit, then
(82)ρ(t)=1,θ(t)=t,E=K.
Hence, the results presented in this section reduces to the results presented in [2] for Theodorsen’s integral equation.
5. Discretizing (47)
In this paper, we will discretize (47) instead of (46). The numerical method used here is based on strict discretization of the integrals in the operator E by the trapezoidal rule which gives accurate results since the integrals are over 2π-periodic. Let n be a given even positive integer. We define n equidistant collocation points si in the interval [0,2π] by
(83)ti:=(i-1)2πn,i=1,2,…,n.
Then, for 2π-periodic function γ(t), the trapezoidal rule approximates the integral
(84)I=∫02πγ(t)dt
by
(85)In=2πn∑i=1nγ(ti).
If the function γ(t) is continuous, then |I-In|→0. If the integrand γ(t) is k times continuously differentiable, then the rate of convergence of the trapezoidal rule is O(1/nk). For analytic γ(t), the rate of convergence is better than O(1/nk) for any positive integer k [7, page 83]. See also [8].
For γ∈H, the integral operator N will be discretized by the Nyström method as follows:
(86)Nnγ(s)=2πn∑j=1nN(s,tj)γ(tj).
Hence, we have
(87)∥(N-Nn)γ∥∞=maxs∈[0,2π]|∫02πN(s,t)γ(t)dt-2πn∑j=1nN(s,tj)γ(tj)|.
Since the kernel N(s,t) is continuous on both variables and since the function γ(t) is continuous, we have [9]
(88)∥(N-Nn)γ∥∞⟶0.
The integral operator M1 will be discretized by the Nyström method as follows:
(89)M1,nγ(s)=2πn∑j=1nM1(s,tj)[γ(tj)-γ(s)].
Since the kernel M1(s,t) is continuous on both variables and since the function γ(t) is continuous, we have [9]
(90)∥(M1-N1,n)γ∥∞=maxs∈[0,2π]|∑j=1n∫02πM1(s,t)γ(t)dtnnnnnnnnnnnnn-2πn∑j=1nM1(s,tj)γ(tj)|⟶0.
To discretize the operator Kγ(s), we first approximate the function γ(s) by the interpolating trigonometric polynomial of degree n/2 which interpolates γ(s) at the n points tj, j=1,2,…,n. That is,
(91)γ(s)≈∑i=0n/2aicosis+∑i=0n/2-1bisinis.
Then Kγ(s) is approximated by
(92)Knγ(s)=∑i=1n/2aisinis-∑i=0n/2-1bicosis,
where [6]
(93)∥(K-Kn)γ∥∞⟶0.
The integral operator M is then discretized by
(94)Mn=M1,n-Kn.
Then, it follows from (90) and (93) that
(95)∥(M-Mn)γ∥∞⟶0.
The operator Mn is bounded operator since the operator M1,n is bounded (M1(s,t) is continuous) and the operator Kn is bounded operator (see [6]).
Since the kernel N(s,t) is continuous and λ=1 is not an eigenvalue of the kernel N(s,t) [1], the operators I-Nn are invertible and (I-Nn)-1 are uniformly bounded for sufficiently large n [9]. Hence, we discretize the operator E by the bounded operator
(96)En:=-(I-Nn)-1Mn.
Lemma 17.
If γ∈H, then
(97)∥(E-En)γ∥∞⟶0.
Proof.
Let ϕ:=Eγ and ϕn:=Enγ, then
(98)(I-N)ϕ=-Mγ,(I-Nn)ϕn=-Mnγ.
Let also ϕ^n be the unique solution of the discretized equation
(99)(I-Nn)ϕ^n=-Mγ.
Thus, we have
(100)∥(E-En)γ∥∞=∥ϕ-ϕn∥∞≤∥ϕ-ϕ^n∥∞+∥ϕ^n-ϕn∥∞.
Since the kernel N is continuous and Nn is the discretization of N, then it follows from [9, page 108] that
(101)∥ϕ-ϕ^n∥∞⟶0.
Since (I-Nn)-1 is bounded and γ is continuous, then (95) implies that
(102)∥ϕ^n-ϕn∥∞=∥(I-Nn)-1(M-Mn)γ∥∞≤∥(I-Nn)-1∥∞∥(M-Mn)γ∥∞⟶0,
which with (100) and (101) implies (97).
To calculate the function Sk in (47) for a given Sk-1, we replace the operator E in (47) by the approximate operator En to obtain
(103)Sk,n(s)-s=EnlnR(Sk-1(·))ρ(·)(s),
where Sk,n is an approximation to Sk. Substituting s=ti and i=1,2,…,n, in (103) we obtain
(104)Sk,n(ti)-ti=EnlnR(Sk-1(·))ρ(·)(ti),i=1,2…,n.
Equation (104) can be rewritten as
(105)(I-Nn)[Sk,n(ti)-ti]=-MnlnR(Sk-1(·))ρ(·)(ti),dsasfaafli=1,2,…,m
which represents an n×n linear system for the unknown Sk,n(t1),Sk,n(t2),…,Sk,n(tn). By obtaining Sk,n(ti) for i=1,2…,n, the function Sk,n(s) can be calculated for s∈[0,2π] by the Nyström interpolating formula. In the following lemma, we prove the uniform convergence of the approximate solution Sk,n of discretized equation (103) to the solution Sk of (46).
Lemma 18.
Consider
(106)∥Sk,n-Sk∥∞⟶0asn⟶∞.
Proof.
Let γ(s):=ln(R(Sk-1(s)))/ρ(s). Then, we have
(107)Sk,n-Sk=Enγ-Eγ=(En-E)γ.
Hence,
(108)∥Sk,n-Sk∥∞≤∥(E-En)γn∥∞.
The lemma is then followed from (97).
The proof of the uniform convergence of the approximate solution Sk,n to the boundary correspondence function S is given in the following theorem.
Theorem 19.
If ε<1, then
(109)∥Sk,n-S∥∞⟶0ask,n⟶∞.
Proof.
We have
(110)∥Sk,n-S∥∞≤∥Sk,n-Sk∥∞+∥Sk-S∥∞.
Since ε<1, it follows from (77) that ∥Sk-S∥∞→0 as k→∞. The theorem is then followed from (106).
6. The Algebraic System
Let t be the n×1 vector t:=(t1,t2,…,tn)T where T denotes transposition. Then, for any function γ(t) defined on [0,2π], we define γ(t) as the n×1 vector obtained by componentwise evaluation of the function γ(t) at the points ti, i=1,2,…,n. As in MATLAB, for any two vectors x and y, we define x·*y as the componentwise vector product of x and y. If yj≠0, for all j=1,2…,(m+1)n, we define x·/y as the componentwise vector division of x by y. For simplicity, we denote x·*y by xy and x·/y by x/y.
Let xk-1=Sk-1(t)-t (given) and xk=Sk,n(t)-t (unknown). Then system (105) can be rewritten as
(111)(I-B)xk=-ClnR(xk-1+t)ρ(t),k=1,2,…,
where I is the n×n identity matrix, B is the discretized matrix of the operator N, and C is the discretized matrix of the operator M [1]. Linear system (111) is uniquely solvable [4, 10, 11].
We start the iteration in (47) with Sk(t)=t and iterate until ∥Sk-Sk-1∥∞<tol where tol is a given tolerance; that is, we start the iteration in (111) with x0=0 and iterate until ∥xk-xk-1∥∞<tol. Each iteration in (111) requires solving a linear system for xk given xk-1. Linear system (111) is solved in O(nlnn) operations by the fast method presented in [11, 12] which is based on a combination of the MATLAB function gmres and the MATLAB function zfmm2dpart in the MATLAB toolbox FMMLIB2D [13]. In the numerical results below, for function zfmm2dpart, we assume that iprec =4 which means that the tolerance of the FMM is 0.5×10-12. For the function gmres, we choose the parameters restart =10, gmrestol =10-12, and maxit =10, which means that the GMRES method is restarted every 10 inner iterations, the tolerance of the GMRES method is 10-12, and the maximum number of outer iterations of GMRES method is 10. See [11, 12] for more details.
By obtaining xk, we obtain the values Sk,n(ti) for i=1,2,…,n. Then, the function Sk,n(s)-s can be calculated for s∈[0,2π] by the Nyström interpolating formula. The convergence of Sk,n(s) to S(s) follows from Theorem 19. Then the values of the mapping function Ψ can be computed from (2). The interior values of the mapping function can be computed by the Cauchy integral formula which can be computed using the fast method presented in [12].
7. Numerical Examples
In this section, we will compute the conformal mapping from three simply connected regions G1, G2, and G3 onto three simply connected regions Ω1, Ω2, and Ω3. The boundaries Γ1, Γ2, and Γ3 of the regions G1, G2, and G3 are parameterized by
(112)η(t)=ρ(t)eit,0≤t≤2π,
where the function ρ(t) is given by
(113)Γ1:ρ(t)=1+14cos34t,Γ2:ρ(t)=1+34cos4t,Γ3:ρ(t)=ecostcos22t+esintsin22t.
The boundaries L1, L2, and L3 of the regions Ω1, Ω2, and Ω3 are parameterized by
(114)η(t)=R(t)eit,0≤t≤2π,
where the function R(t) is given by
(115)L1:R(t)=1,L2:R(t)=α1-(1-α2)cos2t,nnnnnα=0.6180339630899485,L3:R(t)=1+110cos8t.
The curves L1, L2, and L3 satisfy the ε-condition where
(116)L1:ε=∥R′(t)R(t)∥∞=0,L2:ε=∥R′(t)R(t)∥∞=0.5<1,L3:ε=∥R′(t)R(t)∥∞=0.80403<1.
The numerical results obtained with n=4096 and tol=10-12 are shown in Figures 1, 2, and 3. The error norm ∥xk-xk-1∥∞ versus the iteration number k in (111) is shown in Figure 4. It is clear from Figure 4 that the number of iterations in (111) depends only on the boundary L of the image region. More precisely, it depends on ε. The iterations in (111) converge only if ε<1. For small ε, a few number of iterations are required for convergence. For values of ε close to 1, a large number of iterations are required for convergence.
The conformal mappings from G1 onto Ω1, Ω2, and Ω3.
The conformal mappings from G2 onto Ω1, Ω2, and Ω3.
The conformal mappings from G3 onto Ω1, Ω2, and Ω3.
The error norm ∥xk-xk-1∥∞ versus the iteration number k for (a) the conformal mapping from G1 onto Ω1, Ω2, and Ω3, (b) the conformal mapping from G2 onto Ω1, Ω2, and Ω3, and (c) the conformal mapping from G3 onto Ω1, Ω2, and Ω3.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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