Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

The spectral problem for a class of highly oscillatory Fredholm integral operators

Language

English

Abstract

Let ℱω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)e iωx-y. We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;ω) to the associated Fredholm integral equation f = μℱωf + a as ω→∞, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

Keywords

Asymptotic behaviour of highly oscillatory solutions, Asymptotic behaviour of spectrum, Complex-symmetric Fredholm integral operator, Fredholm integral equations, Highly oscillatory kernel

Publication Date

2010

Source Publication Title

IMA Journal of Numerical Analysis

Volume

30

Issue

1

Start Page

108

End Page

130

Publisher

Oxford University Press

DOI

10.1093/imanum/drn060

Link to Publisher's Edition

http://dx.doi.org/10.1093/imanum/drn060

ISSN (print)

02724979

ISSN (electronic)

14643642

This document is currently not available here.

Share

COinS