ON THE PRANDTL EQUATION ON A FINITE INTERVAL

D. Kapanadze

I.Javakhishvili Tbilisi State University A.Razmadze Mathematical Institute of the Georgian Academy of Sciences

The purpose of this paper is the investigation of the basic integro-differential equation of a finite span aeroplane wing

−Γ(z) = πV b(z) [\frac{α(z)}{2} + Q(z) + \frac{1}{4πV} ∫_{−l}^{+l}\frac{Γ′(t)}{z − t}dt]  (1)

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 Γ(z) represents circulation of the wing, while functions b(z), α(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

Γ(−l) = Γ(+l) = 0.  (2)

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 H H_{p}^{s}, s ∈ IR, 1 < p < +∞, namely Γ ∈ H̃ H_{p}^{s}([−l, +l]) and b[α/2 + Q] ∈ H H_{p}^{s−1}([−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.

Theorem 1. Let s ∈ IR and

s = [s] + {s}, [s] = 0, ±1, ±2, ..., 0 ≤ {s} < 1

be the decomposition of s into integer and fractional parts. The operator A

Aφ(x) = c_{0}(x)φ(x) + c_{1} \frac{1}{πi}∫_{−l}^{+l}\frac{φ′(t)}{t − x}dt, c_{1} = const

A : H̃ H_{p}^{s}([−l, +l]) ↦ H H_{p}^{s−1}([−l, +l])

is Fredholm if and only if

|\{s\} − \frac{1}{p}| ≠ \frac{1}{2}.

32 Bulletin of TICMI

If this condition holds, the operator A is invertible, is invertible from the left or is invertible from the right provided σ iz zero, positive or negative, respectorly. Here

σ = [s] if | \{s\} − \frac{1}{p} |< \frac{1}{2},

σ = [s] + 1 if \{s\} − \frac{1}{p} > \frac{1}{2},

σ = [s] − 1 if \{s\} − \frac{1}{p} < −\frac{1}{2},

and IndA = −σ.

R e f e r e n c e s

[1] Belotserkovski C., Lifanov I., The Numerical Methods in Singular Integral Equations. Nauka, Moscow, 1985 (Russian).

[2] Golubev V., Lectures on Airfoil Wing. Gostechizdat, Moscow, 1949 (Russian).

[3] Duduchava R., On Multidimensional Singular Integral Operators. I-II, J. Operator Theory, 11, 1984, 41-76, 199-214.

[4] Duduchava R., Integral Equations with Fixed Singularities. Teubner, Leipzig, 1979.

[5] Triebel R., Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978.

[6] Weissinger R., U^{..}ber Eine Erweiterung der Prandtlschen Theorie der Tragenden Linie. Mathem. Nachrichten, 2, H.1/2, 1949, 46-106.