Eigenfunctions of the Airyʼs integral transform: Properties, numerical computations and asymptotic behaviors

This paper is devoted to the study of the spectrum of the integral operator with Airyʼs kernel. We provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator. The first method is based on the use of a differential operator which commutes with...

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Bibliographic Details
Published in:	Journal of mathematical analysis and applications Vol. 389; no. 2; pp. 989 - 1005
Main Authors:	Karoui, Abderrazek, Mehrzi, Issam, Moumni, Taher
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 15.05.2012
Subjects:	Airyʼs function Airyʼs integral transform Asymptotic behaviors Eigenfunctions and eigenvalues Gaussian quadrature method WKB method Airyʼs integral transform Asymptotic behaviors Gaussian quadrature method Airyʼs function Eigenfunctions and eigenvalues WKB method
ISSN:	0022-247X, 1096-0813
Online Access:	Get full text
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Summary:	This paper is devoted to the study of the spectrum of the integral operator with Airyʼs kernel. We provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator. The first method is based on the use of a differential operator which commutes with this integral operator. The second method is based on a Gaussian quadrature method applied to the finite Airyʼs transform. The asymptotic behavior of the eigenfunctions is studied by the use of a WKB method. Some numerical examples are given to illustrate results of this work.
ISSN:	0022-247X 1096-0813
DOI:	10.1016/j.jmaa.2011.12.036