Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations
type: Item
instance of: scholarly article
title: Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations
zbMATH Open document ID: 1155.35008
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2008-09-12T00:00:00Z
review text: The author introduces functions \(w_-(x)=Ai(x)Bi(x)-Ai^2(x)\) and \(w_+(x)=Ai(x)Bi(x)+Ai^2(x),\) where \(Ai(x), /, Bi(x)\) are Airy functions of the first and second kind, respectively. He proves that half integral of the Airy functions \(D^{-1/2}Ai(x)\) and \(D^{-1/2}Gi(x)\) (\(Gi(x)\) is the Scorer function) can be expressed in the terms of \(w_-(x)\) and \(w_+(x).\) Based on that the author establishes general formulas for computing semi-integral derivatives of \(Ai(x)\) and \(Gi(x)\). Here the functional derivatives of order \(s>-1\) of the function \(f(x)\) defined by \(D^sf(x)=\frac{1}{2\pi}\int_{-\infty}^\infty | \xi| ^s \hat{f}(\xi)e^{i \xi x} d\xi,\) where \(\hat{f}(\xi)\) is the Fourier transform of the function \(f(x)\). Some applications to Korteweg-de Vries type equations and for the Ostrovsky equation are given.
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 33E20
zbMATH DE Number: 5343367
zbMATH Keywords: Airy functions
zbMATH Keywords: Scorer function
zbMATH Keywords: fractional derivatives
zbMATH Keywords: fractional integrals
zbMATH Keywords: Korteweg-de Vries type equations
author: V. V. Varlamov
reviewed by: Vladimir L. Makarov
MaRDI profile type: Publication
full work available at URL: s00033-007-6074-2
OpenAlex ID: W2033579316
DOI: 10.1007/S00033-007-6074-2
description: scientific article; zbMATH DE number 5343367