Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications
type: Item
instance of: scholarly article
title: Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications
zbMATH Open document ID: 1182.26019
published in: Journal of Computational and Applied Mathematics
publication date: 2009-09-29T00:00:00Z
review text: Author's abtract: Riesz fractional derivatives of a function \(D^{\alpha }_{x}f(x)\) (also called Riesz potentials) are defined as fractional powers of the Laplacian. Asymptotic expansions for large \(x\) are computed for the Riesz fractional derivatives of the Airy function of the first kind \(Ai(x)\) and the Scorer function \(Gi(x)\). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions \(D^{\alpha }_{x}\{Ai(x)Bi(x)\}\) and \(D^{\alpha }_{x}\{Ai^2(x)\}\), via \(D^{\alpha }_{x}Ai(x)\) and \(D^{\alpha }_{x}Gi(x)\). Here, \(Bi(x)\) is the Airy function of the second type. Integral representations are presented for the function \(A_{2}(a,b;x) = Ai(x - a)Ai(x - b)\) with \(a, b \in \mathbb R\) and its Hilbert transform. Combined with the above asymptotic expansions, they can be used for computing asymptotics of the Hankel transform of \(D^{\alpha }_{x}\{A_2 (a, b; x)\}\). These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 5608819
zbMATH Keywords: Riesz fractional derivatives
zbMATH Keywords: Airy functions
zbMATH Keywords: scorer functions
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: Ostrovsky equation
author: Nico M. Temme
author: V. V. Varlamov
reviewed by: Ravinder Krishna Raina
MaRDI profile type: Publication
full work available at URL: j.cam.2009.05.029
OpenAlex ID: W2079026301
DOI: 10.1016/J.CAM.2009.05.029
description: scientific article; zbMATH DE number 5608819