The numerical solution of the generalized airfoil equation
type: Item
instance of: scholarly article
title: The numerical solution of the generalized airfoil equation
zbMATH Open document ID: 0762.65091
author: Dietmar Berthold
author: W. Hoppe
author: B. Silbermann
published in: Journal of Integral Equations and Applications
publication date: 1993-01-26T00:00:00Z
review text: The authors consider the singular integral equation \[ {1\over\pi} \int_{-1}^ 1 {u(y) \over {y-x}}dy-{c\over y} \int_{-1}^ 1 \ln| y-x| u(y)dy+\int_{-1}^ 1 k(x,y)u(y)dy=f(x), \qquad x\in(-1,1),\tag{1} \] where \(f\) and \(k\) are given Hölder continuous functions, \(c\) is a complex number, and \(u\) is the unknown function. Using some mapping properties of the operator with logarithmic kernel, they establish a quadrature method for the approximate solution of equation (1), where the discretization of the first two integral operators in (1) can be done without any integration. The paper contains stability and convergence results for the quadrature method and discusses iterative two-grid and multi-grid methods for determining an approximate solution to equation (1) with a computational complexity of \(O(n^ 2)\).
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E05
Mathematics Subject Classification ID: 76B10
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 86118
zbMATH Keywords: generalized airfoil equation
zbMATH Keywords: singular integral equation
zbMATH Keywords: logarithmic kernel
zbMATH Keywords: quadrature method
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: iterative two-grid and multi-grid methods
zbMATH Keywords: computational complexity
reviewed by: Johannes Elschner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075695
OpenAlex ID: W2084125200
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DOI: 10.1216/JIEA/1181075695
description: scientific article; zbMATH DE number 86118