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Zbl 1155.35008
Varlamov, Vladimir
Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations. (English)
[J] Z. Angew. Math. Phys. 59, No. 3, 381-399 (2008). ISSN 0044-2275; ISSN 1420-9039/e

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 $D^sf(x)=\frac{1}{2\pi}\int_{-\infty}^\infty \vert \xi\vert ^s \hat{f}(\xi)e^{i \xi x} d\xi,$ 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.
[Vladimir L. Makarov (Ky{\"\i}v)]

MSC 2000:: *35C15 Integral representations of solutions of PDE
35Q53 KdV-like equations
33E20 Functions defined by series and integrals

Keywords: Airy functions; Scorer function; fractional derivatives; fractional integrals; Korteweg-de Vries type equations

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Zentralblatt MATH Berlin [Germany]