Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle

I. S. Polyanskii ; K. O. Loginov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 1, pp. 107-129

Résumé

In the article, an approximate analytical solution of the problem of conformal mapping of internal points of an arbitrary polygon to a unit circle is developed. At the preliminary stage, the conformal mapping problem is formulated as a boundary value problem (Schwartz problem). The latter is reduced to the solution of the Fredholm integral equation of the second kind with a Cauchy-type kernel with respect to an unknown complex density function at the boundary domain, followed by the calculation of the Cauchy integral. The developed approximate analytical solution is based on the Cauchy kernel decomposition in the Legendre polynomial system of the first and second kind. A priori and a posteriori estimates of the convergence and accuracy of the given solution are fulfilled. The exponential convergence of the solution in $L_2\left([0,1]\right)$ and the polynomial one in $C\left([0,1]\right)$ are defined. Calculations on test examples are given for a visual comparison of the effectiveness of the developed solution.

Détail

Keywords: conformal mapping, arbitrary polygon, Schwartz problem, logarithmic double layer potential, complex density function, Fredholm equation
Mots-clés : Legendre polynomials.