Computation of the spectra of some integral operators and application to the numerical solution of some linear integral equations

In this paper, we present two methods of computing the spectrum of a compact integral operator. The first method is based on an exact matrix representation of the operator. The second method uses a convenient quadrature method to discretisize the integral operator and to provide accurate approximati...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 218; no. 7; pp. 3217 - 3229
Main Author: Aouicha, Hichem Ben
Format: Journal Article
Language:English
Published: Elsevier Inc 01.12.2011
Subjects:
Approximation
Computation
Eigenvalues and eigenfunctions
Gaussian quadrature
Hilbert–Schmidt operators
Integral equations
Integral equations of the first kind
Integrals
Mathematical analysis
Mathematical models
Matrix representation
Operators
Spectra
Eigenvalues and eigenfunctions
Matrix representation
Gaussian quadrature
Integral equations of the first kind
Hilbert–Schmidt operators
ISSN:0096-3003, 1873-5649
Online Access:Get full text
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Summary:In this paper, we present two methods of computing the spectrum of a compact integral operator. The first method is based on an exact matrix representation of the operator. The second method uses a convenient quadrature method to discretisize the integral operator and to provide accurate approximations to the spectrum and the eigenfunctions of this later. Also, we show how our methods can be used in the framework of some stable procedures for the approximation of f † the normal solution of the minimal L 2-norm of the integral equation of the first kind Af = g, which is often an ill-posed equation. These procedures are based on a spectral expansion of the operator A. To finish, we give some numerical examples that illustrate the results of this work.
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ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2011.08.062