Fast numerical schemes for Fredholm integral equations of the second kind

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" Fast numerical schemes for Fredholm integral equations of the second kind "
S. Xia I. Koltracht

Document Type	:	Latin Dissertation
Language of Document	:	English
Record Number	:	1113500
Doc. No	:	TLpq304445194
Main Entry	:	I. Koltracht
	:	S. Xia
Title & Author	:	Fast numerical schemes for Fredholm integral equations of the second kind\ S. XiaI. Koltracht
College	:	University of Connecticut
Date	:	1998
student score	:	1998
Degree	:	Ph.D.
Page No	:	107
Abstract	:	Fast numerical schemes for Fredholm integral equations of the second kind have been developed. The integral equations are firstly discretized by the open Clenshaw-Curtis quadrature rule on a nodal point set usd\Xi\sb{n}usd. Generally n has to be chosen fairly large in order to obtain a certain accuracy. When the kernels are sufficiently smooth, we have shown that the linear systems of equations can be approximated well by their low rank approximations on usd\Xi\sb{m}usd with usdm \ll nusd by the eigenvalue expansions or the singular value decompositions of the integral operators. Most computations are now accomplished on usd\Xi\sb{m}usd. The Chebyshev expansions are used to define the interpolation formulas. We have shown that, if the kernel usd\kappa\ \in\ C\sp{p}usd and the right hand side function usdy\ \in\ C\sp{q}usd for some integers usdp,q > 0usd, the schemes converge at the rate of usdo(1/m\sp{p-l}) + o (1/n\sp{\nu -1}usd), where the integer usd\nu \geusd min(p,q). When the kernels usd\kappa(s,tusd) are non-smooth along the line usds=tusd, we have firstly described a high order quadrature rule. Then, we proposed two iteration methods to efficiently solve the corresponding equations.
Subject	:	Chebyshev expansions
	:	Fredholm equations
	:	Mathematics
	:	Pure sciences
	:	Quadrature