A Boundary Integral Equation Method for Computing Numerical Conformal Mappings onto the Disk with Rectilinear Slit and Spiral Slits Regions

1. INTRODUCTION

The identification of canonical regions plays a crucial role in conformal mappings of multiply connected regions. The regions identified as canonical include the disk with circular slits, the annulus with circular slits, the circular slit region, the radial slit region, and the parallel slit region. Furthermore, additional canonical regions for conformal mappings include the disk with spiral slits region, annulus with spiral slits region, spiral slits region, and straight slits region [1]-[13]. Nasser’s method of computing conformal mapping is based on Riemann-Hilbert problem [2], [5], [14], while Sangawi’s methods rely on integral equations satisfy the interior non-homogeneous boundary relationship [8]-[12].

The canonical slit regions introduced by Koebe [1], DeLillo et al. [15], and Nasser [5] are special cases of the spiral slits region. Sangawi [9]-[11] and Sangawi et al. [12] have demonstrated conformal mapping of bounded multiply connected regions onto the second, third, and fourth categories of Koebe’s canonical slit regions using a boundary integral equation method. In Nasser [14] study, the study of bounded multiply connected region onto the disk with rectilinear slit and spiral slits region was facilitated by reformulating the conformal mapping as a Riemann-Hilbert problem. The present paper aims to establish a new boundary integral equation method for numerical conformal mappings from Ω onto Ω₁ and Ω₂.

The design of the study is as follows: Section 2 presents some necessary materials. A derivation of integral equation method for computing the function 𝓕 has been presented in Section 3. The boundary integral equation method has been illustrated through examples provided in Section 4. Lastly, Section 5 comprises of the conclusion.

2. NECESSARY MATERIALS

A bounded multiply connected region Ω of connectivity M + 1. The boundary Γ consists of M + 1 smooth Jordan curves Γ_ι,ι=0,1,…,M as demonstrated in the following, (see Fig. 1)

Fig. 1. Mapping of Ω onto Ω₁ and Ω₂.

The curve Γ_ι is parametrized by a 2π periodic twice continuously differentiable complex function ξ_ι (t)

The complete parameter I is the combination of M + 1 disjoint intervals I_ι, ι = 0,…,M. The entire boundary Γ on I is defined by parametrization ξ (t)

Assuming is a complex function that is continuously differentiable with a periodicity of 2π ∀t∈I_ι. The generalized Neumann kernel that is formed using can be described as[16]:

The classical Neumann kernel is the generalized Neumann kernel formed with , i.e.

The adjoint kernel N* (s,t) of the Neumann kernel is as follows:

The generalized Neumann kernel Ñ(s, t) is as follows:

Refer to Sangawi [10] for the definitions of N, Ñ and N^*.

3. INTEGRAL EQUATION METHOD FOR COMPUTING THE FUNCTION 𝓕

The canonical region can be described as a disk with a finite straight slit along the line where Im 𝓕 =0, as well as M−1 finite spiral slits. Additionally, there is a rectilinear slit, which refers to a slit that lies on a straight line.

The variable α represents the angles of intersection between the line and the real axis. There is also a spiral slit, which refers to a slit that is located on a logarithmic spiral.

Im

Where the oblique angles α are prescribed in advance.

Assume that the function 𝓕(ζ) maps the curve Γ₀ onto the circle with radius e^-R₀, the circle Γ₁ onto a finite rectilinear slit that lies on the line where Im (𝓕(ζ))=0, and the curves Γ_ι, where ι=2,…,M, onto M−1 finite spiral slits with oblique angles θ_ι,ι=2,…,M. Therefore, the mapping function that transforms Ω onto Ω₁ and Ω₂ fulfills the following conditions.

The values R₀,R₁,…,R_M are real constants that have not been determined, , R(t)=(R₀,0,R₂,…,R_M) [14]. Hence 𝓕(ζ) satisfy

And 𝓕(ζ) can be reformulated as [14]:

Where,

ρ is a radius of Γ₁ c=1 for Ω₁, c=0 for Ω₂, ĥ (ξ) is an analytic function in Ω₁ for c=1 and ĥ (ξ) is an analytic function in Ω₂ for c=0. And then define S(t) by,

We assume that,

which implies that,

After some algebraic manipulations, we obtain,

From the definitions of F(ζ) and 𝓕(ζ) we obtain,

Let,

Combining (9) and (11), we obtain,

Equation (10), yields,

We reach the following from (7):

then

where

By obtaining h_ι, ι=0,…,M, from second equation in Theorem 2; Sangawi [11] we obtain

By using Theorem 1 in Sangawi [11], (11), (12) and after some algebraic operations we achieve the following:

Assume that ξ=ξ (t) and ξ=ξ (s). Then by placing in (17), we realize

this can be written in its operator form (Ñ=−N*)

As a result, (18) is not uniquely solvable. To deal with this problem, observe

which implies

The combination of (18) and (19) gives:

In the light of [14, Theorem 2], (20) is uniquely solvable. gives the value of , by using the following equation.

where υ_ι is a real constant integration, we see that,

h_ι is obtained through solving (13) and (11) in Sangawi [11] from which r_ι is provided through (16). Having solved (20) we are granted the value . We obtain υ_ι through the equations (37), (38) and (12) in Sangawi [11] from which is acquired, after that 𝓕(ζ) is attained by,

Using the Cauchy integral formula, the interior value of 𝓕(ζ) is determined.

4. NUMERICAL EXAMPLES

Nyström’s method alongside the trapezoidal rule [17] [18] was used to solve (13) in Sangawi [11] and (20). The computational details are almost identical to [19], [20].

Some test regions of connectivity three, four, and seven have been used for numerical experiments [14]. MATLAB R2020a was used to carry out all the computations. In each boundary Γ_j the same number of collocation point has been used. Ω, Ω₁ and Ω₂ are shown in Figures 2-4. Tables 1-3 exhibit our computed values of r_ι,ι=0,…,6 compared to those of Nasser [14].

Fig. 2. Mapping Ω onto Ω₁ and Ω₂ with three connectivity.

Fig. 3. Mapping Ω onto Ω₁ and Ω₂ with four connectivity.

Fig. 4. Mapping Ω onto Ω₁ and Ω₂ with seven connectivity.

Table 1: Computed values of r_ι,ι=0,1,2

Table 2: Computed values of r_ι,ι=0,…,3 with n=128

Table 3: Computed values of r_ι,ι=0,…,6

Example 1

The region with the following boundaries:

Table 1 give the computed values of r and Ω, Ω₁ and Ω₂ are shown in Figure 2.

Example 2

The region Ω bounded by four multiply connected region, Nasser [14],

The values of r_ι,ι=0,…,3 are listed in Table 2 and Ω, Ω₁ and Ω₂ are shown in Figure 3.

Example 3

The region Ω bounded by seven multiply connected region, Nasser [11],

The values of r_ι,ι=0,…,6 are listed in Table 3 and Ω, Ω₁ and Ω₂ are shown in Figure 4. Some more examples are shown in Fig. 5.

Fig. 5. Ω, Ω₁ and Ω₂ with high connectivity.

5. CONCLUSIONS

The present study proposes a new boundary integral equation for the conformal mapping of multiply connected regions onto the disk with rectilinear slit and spiral slits regions, Ω₁ and Ω₂. We used the proposed method to compute several mappings of some test regions and computed the boundary values of the mapping function. The interior mapping function was then determined using Cauchy’s integral formula. Numerical examples were provided to demonstrate the high accuracy of the boundary integral equation method.