The approximate solution of 2D Dirichlet problem in doubly connected domains
type: Item
instance of: scholarly article
title: The approximate solution of 2D Dirichlet problem in doubly connected domains
zbMATH Open document ID: 1416.65506
DOI: 10.1155/2018/6951513
author: E. A. Shirokova
author: Atallah El-Shenawy
published in: Advances in Mathematical Physics
publication date: 2018-12-11T00:00:00Z
review text: The authors describe a method for solving the Laplace equation subject to Dirichlet boundary conditions in a doubly connected domain \(\Omega \subset\mathbb{R}^2\) with \(0 \notin \Omega\). The starting point is to write the solution \(u\) in the form \(u(x,y)=\Re(B(z))+A\log(z)\), \(z=x+iy\), where \(B\) is analytic in \(\Omega\) and \(A\) is a real constant. \(B\) is represented using the Cauchy integral formula. The boundary of \(\Omega\) is given in parametric form and approximated by truncated Fourier polynomials. The determination of \(B\) on \(\partial\Omega\) results in solving two Fredholm integral equation of the second kind. Their solutions are also approximated by truncated Fourier series. At the end, a linear system for the unknown coefficients in the Fourier series is left to solve. Finally, \(A\) can be determined by a least square method. The numerical solution is improved for points near \(\partial\Omega\) in this way taking the singularity of the kernel in the Cauchy integral into account. Some numerical experiments are reported.
reviewed by: Rolf Dieter Grigorieff
Mathematics Subject Classification ID: 65N99
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 6991999
zbMATH Keywords: Laplace equation
zbMATH Keywords: doubly connected domain
zbMATH Keywords: Fredholm integral equation
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Cauchy integral
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/6951513
OpenAlex ID: W2832206975
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Wikidata QID: Q129490438
description: scientific article; zbMATH DE number 6991999