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Conformal Mapping of Right Circular Quadrilaterals

Vladislav V. Kravchenko, R. Michael Porter

Abstract

We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t,lambda of the mapping problem. The parameter lambda is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm-Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter lambda. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series (SPPS) method for numerical solution. Rate of convergence, accuracy, and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favorably with previously known methods for this type of problem.

Conformal Mapping of Right Circular Quadrilaterals

Abstract

We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t,lambda of the mapping problem. The parameter lambda is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm-Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter lambda. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series (SPPS) method for numerical solution. Rate of convergence, accuracy, and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favorably with previously known methods for this type of problem.

Paper Structure

This paper contains 11 sections, 2 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Sturm-Liouville equation
  3. Boundary Value Problem
  4. The Degenerate S.C.Q.s
  5. Solution by Iterated Integrals
  6. Variation of Curvature
  7. Algorithm for the One-Parameter Problem
  8. Two-Parameter Problem
  9. Domain of univalence.
  10. Numerical Results
  11. Conclusions

Key Result

Proposition 4.1

KP Let $\psi_0$ and $\psi_1$ be given, and suppose that $y_\infty$ is a function which does not vanish and which satisfies the ordinary differential equation on $[0,1]$. Choose $q_0=1/y_\infty^2$, $q_1=\psi_1\,y_\infty^2$ and define $X^{(n)}$, $\widetilde{X}^{(n)}$ to be the iterated integrals determined by $(q_0,q_1)$ and by $(q_1,q_0)$ respectively. Then for every $\lambda\in\hbox{C}$ the funct

Figures (7)

  • Figure 1: Graph of $\kappa$ as a function of $\lambda$ for $t=0.1\pi, 0.2\pi,0.3\pi,0.4\pi$.
  • Figure 2: Images $f(\hbox{D})$ for $t=\pi/4$ and $\lambda=1.3,\ 1.4,\ 1.5$. The images are depicted near the point $w=0.5$, where it is perceived that the rightmost figure is a non-schlicht region.
  • Figure 3: Images $f(\hbox{D})$ for $t=\pi/4$ and (a) $\lambda=0=\lambda_\infty=0$, (b) $\lambda=-0.32219$, $\lambda=-0.91570$ and (d) $\lambda=-1.43554$ (schematic drawing of boundary). The value in (d) corresponds to second largest zero of $\kappa(\lambda)$, after $\lambda_\infty$. The values for (b), (c) were chosen to give equal values $\kappa=0.8$.
  • Figure 4: Normalized curvature $\kappa_1$ as a function of $\lambda$ for $t=0.3\pi$. As $\lambda$ passes through the local maxima at height $\kappa_1=2$, the boundary image $f(\partial\hbox{D})$ exhibits the behavior shown in Figure \ref{['fig:neglambda']}(d).
  • Figure 5: Increasing conformal module of $P$ corresponding to increasing $p_2$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 4.1
  • Lemma 7.1