عنوان

Fast numerical schemes for Fredholm integral equations of the second kind

TLpq304445194

انگلیسی

Fast numerical schemes for Fredholm integral equations of the second kind

[Thesis]

S. Xia

I. Koltracht

University of Connecticut

1998

107

Ph.D.

University of Connecticut

1998

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.

Chebyshev expansions

Fredholm equations

Mathematics

Pure sciences

Quadrature

I. Koltracht

S. Xia

p

[Thesis]

276903

a

Y