Conformal Mapping of Right Circular Quadrilaterals

TL;DR

This paper develops a numerical method based on spectral parameter power series to compute conformal mappings from the unit disk to right-angled circular quadrilaterals, improving efficiency and accuracy over previous methods.

Contribution

It introduces a novel application of the SPPS method to solve the nonlinear Sturm-Liouville boundary value problem in conformal mapping, providing better computational performance.

Findings

The SPPS method achieves high accuracy in mapping computations.

The approach demonstrates favorable convergence and computational efficiency.

Numerical results validate the method's effectiveness for this class of problems.

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.