MPHA Seminar, Blocker 627, 19th October 2007, 1:50pm

Speaker: Vladimir Varlamov, UT - Pan American, Edinburg, TX
Title: Fractional Derivatives of Airy Functions and Applications to
Differential Equations

Airy functions have numerous applications in various branches of
mathematics, physics and engineering. The products of Airy functions
turn out to be special functions in their own right.  Moreover, some
fractional derivatives of Airy functions can be expressed in terms of
the above products and therefore play the role of special
functions. Representations are derived for fractional derivatives of
Airy functions $Ai(x)$ and $Gi(x)$, where $Ai(x)$ is the Airy function
of the first kind and $Gi(x)$ is the Scorer function. Fractional
derivatives are computed for $Ai^2(x)$, $Ai(x)Bi(x)$ and
$Ai(x-a)Ai(x-b)$ with real $a$ and $b$, where $Bi(x)$ is the Airy
function of the second kind. Hilbert transforms of the above
fractional derivatives are computed.

One of the powerful methods of investigating Cauchy problems for
nonlinear evolution equations consists in reducing them to integral
equations. The difficulty of treating these nonlinear equations
consists in the presence of a derivative in the nonlinear term. After
reducing a Cauchy problem to the corresponding integral equation part
of this derivative can be transferred to the fundamental
solution. This brings about the necessity of studying fractional
derivatives of both the fundamental solution and the
nonlinearity. Applications are given to Korteweg-de Vries-type
equations.