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\begin{center}
ON THE PRANDTL EQUATION
ON A FINITE INTERVAL

\vspace*{0.2cm}
{\it D. Kapanadze}

\vspace*{0.2cm}
{\it I.Javakhishvili Tbilisi State University}\\
{\it A.Razmadze Mathematical Institute}\\
{\it of the Georgian Academy of Sciences}\\
\end{center}

\vspace*{0.1cm}
\par
The purpose of this paper is the investigation of the basic integro-differential equation of
a finite span aeroplane wing
\begin{equation}
 -\Gamma(z)=\pi Vb(z)\left[ \frac{\alpha(z)}{2} +Q(z)+\frac{1}{4\pi V}{\mathop\int\limits_{-l}^{+l}}\frac{\Gamma^{\prime}(t)}{z-t}dt \right]
\end{equation}
which is known as the Prandtl equation on a finite interval. Equation (1) relates the wing's 
geometrical characteristics to its aerodynamic properties. The unknown function
$\Gamma(z)$ represents circulation of the wing, while functions $b(z), \alpha(z), Q(z)$
are defined by the construction and represent a chord, curvature of a profile and twist 
of the wing (see [2]). V=const is the speed of a stream at infinity and is usually detected
by the experiments. The circulation satisfies the following boundary conditions
\begin{equation}
\Gamma(-l)=\Gamma(+l)=0.
\end{equation}

The Prandtl equation was investigated mostly by the numerical methods (see
[1], [6]). Our purpose is to give precise theory of the Prandtl equation
in the Bessel potential spaces.
 
We shall treat equation (1) as a pseudodifferantial equation in the Bessel
potential spaces $\bH^{s}_{p},$ $s\in \bR,$ $1<p<+\infty$, namely $\Gamma \in 
\tilde{\bH^{s}_{p}}([-l,+l])$ and $b[\alpha/2+Q] \in \bH^{s-1}_{p}([-l,+l])$
(For the standard notation see [5]). The investigation is based on the results of
R. Duduchava on convolution equations with discontinuous symbols (see [4]) and the local 
principle from [3].

Fredholm criteria and the index formula for equation (1) are found.
\par{\bf Theorem 1.} {\it Let $ s\in \bR $ and
$$
s=[s]+\{s\}, \; \; [s]=0,\pm 1,\pm 2, \; ...,  \; \; 0\leq \{s\}<1
$$
be the decomposition of s into integer and fractional parts. The operator {\bf{A}}
$$
 {\bf{A}} \phi(x)=c_{0}(x)\phi(x)+c_{1}\frac{1}{\pi i}{\mathop\int\limits_{-l}^{+l}}\frac{\phi^{\prime}(t)}{t-x}dt, \; \; \; c_{1}=const
$$
$$
 {\bf{A}} :\tilde{\bH^{s}_{p}}([-l,+l]) \mapsto \bH^{s-1}_{p}([-l,+l])
$$
is Fredholm if and only if 
$$
|\{s\}-\frac{1}{p}| \neq \frac{1}{2}.
$$
If this condition holds, the operator {\bf{A}} is invertible, is invertible from 
the left or is invertible from the right provided $\sigma$ iz zero, positive
or negative, respectorly.
\\ Here 
$$
\sigma=[s]  \; \; \; if \; \; \; \mid \{s\}-\frac{1}{p} \mid < \frac{1}{2},
$$
$$
\sigma=[s]+1 \; \; \; if \; \; \; \{s\}-\frac{1}{p} > \frac{1}{2},
$$
$$
\sigma=[s]-1 \; \; \; if \; \; \; \{s\}-\frac{1}{p} < -\frac{1}{2},
$$
and
$$
Ind {\bf{A}}=-\sigma.
$$
}

\vspace*{0.2cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.1cm}
\end{center}
\par
[1]
Belotserkovski C., Lifanov I., The Numerical Methods in Singular Integral Equations.
Nauka, Moscow, 1985 (Russian).
\par
[2]
Golubev V., Lectures on Airfoil Wing. Gostechizdat, Moscow, 1949 (Russian).
\par
[3]
Duduchava R., On Multidimensional Singular Integral Operators.
 I-II, J. Operator Theory, 11, 1984, 41-76, 199-214.
\par
[4]
Duduchava R., Integral Equations with Fixed Singularities.
 Teubner, Leipzig, 1979.
\par
[5]
Triebel R., Interpolation Theory, Function Spaces, Differential Operators.
North-Holland, Amsterdam, 1978.
\par
[6]
Weissinger R., $\mathop{\rm U}\limits^{..}$ber Eine Erweiterung der Prandtlschen Theorie der Tragenden Linie.
 Mathem. Nachrichten, 2, H.1/2, 1949, 46-106.
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