EXPERIMENT ON NUMERICAL CONFORMAL MAPPING OF UNBOUNDED MULTIPLY CONNECTED DOMAIN IN FUNDAMENTAL SOLUTIONS METHOD

We are concerned with the experiment on numerical conformal mappings. A potentially theoretical scheme in the fundamental solutions method, different from the conventional one, has been recently proposed for numerical conformal mappings of unbounded multiply connected domains. The scheme is based on the asymptotic theorem on extremal weighted polynomials. The scheme has the characteristic called “invariant and dual.” Applying the scheme for typical examples, we will show that the numerical results of high accuracy may be obtained. 2000 Mathematics Subject Classification. 30E10, 41A10, 65E05.

Let D and D denote unbounded multiply connected domains whose boundaries γ and γ consist of Jordan curves γ i and γ i (i = 0(1)m), respectively.
We have proposed in [9] the following scheme of approximations of f (z) where the charge points {z n,i } n i=1 are appropriately chosen interior to γ. When D is {w : |w| > r 0 } with radial cuts m j=1 γ i , we propose the algorithm computing approximations of f (z) as follows. (2b) When α (j) i (i = 0(1)n j , j = 0(1)m) are the solutions of a system of (m + 1)(n + 1) simultaneous linear equations using Dirichlet-Neumann and charge conditions [11,12]: Note that the approximations hold, where θ j is the argument of γ j .
The invariant scheme of approximations has been first shown for the numerical Dirichlet problem by Murota [16,17]. It is physically natural and mathematically reasonable.
The solutions of a system of simultaneous linear equations in Scheme 2.1 are also invariant in the sense that the transformation z → az (a > 0) implies (2.6)

A numerical example. We consider a function
This corresponds to Dirichlet-Neumann problem and easy to check the accuracy of the approximation. We apply Scheme 2.1 to compute the approximations of f (z). The charge points interior to γ 1 and the collocation points on γ 1 are so chosen that The accuracy of the errors is estimated by at the points on γ 1 exp 2πj(i− 1) n + πj n + 6, j = √ −1, i = 1(1)n (3.6) and the images of the points on γ 0 under the function f −1 (w).
By the maximum principle for the analytic functions, it is sufficient that the errors are estimated only on the boundary.
The numerical results are presented for the following cases (under a minor modification of the scheme in order to keep the continuity of the argument).
The errors on γ 1 and γ 0 are as follows:  Furthermore, which shows that (2.5) holds with high accuracy. (3b) The charge distribution with l = 0.5. We show the charges in (2b) interior to γ 1 and γ 0 , respectively as follows:  The errors on γ 1 and γ 0 are as follows:  which shows that (2.5) holds with high accuracy. (3c) The charge distribution with l = 0.25. We show the charges in (2b) interior to γ 1 and γ 0 , respectively as follows: Table 3.9. The charge distribution with n = 13, l = 0.25 on γ 1 . The errors on γ 1 and γ 0 are as follows: Table 3.11. The errors with n = 13, l = 0.25 on γ 1 . which shows that (2.5) holds with high accuracy. The numerical example shows: (3d) The data present the distribution of the charges and errors of the approximations, which are symmetric with respect to the real axis and with high accuracy.

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(3e) When the charges and collocation points are distributed uniformly (see the definition in [6]) as (3.4), the example shows that the approximations would hold with high accuracy.
We have examined the case of odd n. The case of n = 12 is shown in the following (3f).
(3f) The charge distribution with l = 0.25. We show the charges in (2b) interior to γ 1 and γ 0 , respectively as follows: Table 3.13. The charge distribution with n = 12, l = 0.25 on γ 1 . The errors on γ 1 and γ 0 are as follows:  (3.20) which shows that (2.5) holds with high accuracy. We have shown the data exactly for the convenience of the readers in order to follow the numerical experiment in Fortran 90 with double precision.
The numerical calculation has been performed in MsDevf90 (PC9821-NEC).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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