Harmonic shears and numerical conformal mappings

In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLimplementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.

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.

Theoretical scheme on numerical conformal mapping of unbounded multiply connected domain by fundamental solutions method

A potentially theoretical scheme in the fundamental solutions method, different from the conventional one, is proposed for numerical conformal mappings of unbounded multiply connected domains. The scheme is introduced from an algorithm on numerical Dirichlet problem, based on the asymptotic theorem on extremal weighted polynomials. The scheme introduced in this paper has the characteristic called “invariant and dual.”

Uniform convergence of numerical conformal mappings of interior domains in the charge simulation method

The uniform convergence of the approximations by new numerical schemes in the charge simulation method, which have been recently proposed by Inoue (1997), will be studied. The exponential decrease of the errors will also be shown.

Theoretical consideration of charge simulation method for numerical conformal mappings in a ring domain

The charge simulation method has been applied to solve a lot of problems in electrical engineering. However, the principle of the method is not known enough even now. This paper is devoted to giving the theoretical and mathematical base for the charge simulation method of numerical conformal mappings in ring domains. Therefore for example, the uniform, convergence of approximations, the theoretical distribution of charge points, and the charges will be mathematically discussed. An example is shown to help understanding of theoretical considerations.