A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains

Abstract

In this paper we propose a new method for solving the 2D Laplace equation with Dirichlet boundary conditions in simply and doubly connected domains. Here, we apply the numerical algorithm based on truncated Fourier series and reduce the corresponding Fredholm integral equation to a finite system of linear equations.

The solution of a mixed boundary value problem for the Laplace equation in a multiply connected domain

“…Introduction. The article extends the results of [1], where the approximate analytical solution of the Dirichlet problem for the Laplace equation in simply connected and doubly connected domains with smooth boundaries was reduced to systems of linear algebraic equations and the solution had the form of the real part of a Cauchy integral. The Laplace equation arises in different areas, such as electrostatics (where it describes the electrostatic potential), stationary potential incompressible fluid flows, and steady-state heat conduction [12].…”

Continued Fractions and Conformal Mappings for Domains with Angle Points

“…Finally, the advantages of the method presented in [19] are the following: 1) it is devoid of auxiliary constructions, 2) it brings us to the mapping function in a polynomial form. The mapping function is a Taylor polynomial for the unit disk or a Laurent polynomial for the annulus in the case of multiconnected domains [1], [18].…”

Continued Fractions and Conformal Mappings for Domains with Angel Points

“…Finally, the advantages of the method presented in [19] are the following: 1) it is devoid of auxiliary constructions, 2) it brings us to the mapping function in a polynomial form. The mapping function is a Taylor polynomial for the unit disk or a Laurent polynomial for the annulus in the case of multiconnected domains [1], [18]. Let us recall the basic construction steps of the reparametrization method [19].…”