The Newton method for solving the Theodorsen integral equation
type: Item
instance of: scholarly article
title: The Newton method for solving the Theodorsen integral equation
zbMATH Open document ID: 0582.30006
DOI: 10.1016/0377-0427(86)90129-9
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Theodorsen's integral equation for a starlike region given by the polar coordinates (\(\phi\),\(\rho\) (\(\phi)\)) of its boundary is solved by applying Newton's method (in a Banach space). It is shown that the linear integral equation for the Newton corrections can be reduced to a Riemann- Hilbert problem on a disk. This approach, which a posteriori was also found in concealed form in an old Russian paper by \textit{B. A. Vertgejm} [Dokl. Akad. Nauk SSSR 119, 12-14 (1958; Zbl 0080.288)], is likely to become important in numerical conformal mapping since in practice these Riemann-Hilbert problems can be solved very efficiently using FFTs. Under the assumption that \(\rho\) '' is Lipschitz continuous the local quadratic convergence is proved, and under the assumptions \(\rho ''\in L^{\infty}\) and \(\| \rho '/\rho \|_{\infty}<1/3\) the global convergence of the (undiscretized) iteration is established.
Mathematics Subject Classification ID: 30C30
Mathematics Subject Classification ID: 35Q15
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 3931336
zbMATH Keywords: Theodorsen's integral equation
zbMATH Keywords: Newton's method
zbMATH Keywords: Riemann-Hilbert problem
author: Otto Huebner
reviewed by: Martin H. Gutknecht
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90129-9
OpenAlex ID: W2007988178
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description: scientific article; zbMATH DE number 3931336