An integro-differential equation of fourth order is considered which describes the deflection of a plate in supersonic gas flow: \[ \chi ^{2}\left( \frac{\omega ^{\prime \prime }}{\left( 1+\omega ^{\prime ^{2}}\right) ^{\frac{3}{2}}}\right) ^{\prime \prime }+\beta _{0}\omega =kK\left( \omega^{^{\prime }},M,\kappa \right)+\theta \omega ^{\prime \prime }\int_{0}^{1}\left[ \left( 1+\omega ^{\prime ^{2}}\right) ^{\frac{1}{2}}-1\right] dx+\varepsilon _{0}q(x),\tag{1} \] where \(0<x<1\), \(\omega =\omega (x)\) is the slope of the plate, \(\chi ^{2}= \frac{h^{2}}{12(1-\mu ^{2})d^{2}}\), \(T=\frac{qd}{Eh}\), \(k=\frac{p_{0}d}{Eh}\), \(d\) is the width of the plate, \(h\) is the thickness of the plate, \(E\) is the Young modulus, \(\mu \) is the Poisson coefficient, \(M\) is the Mach number, \(K\) is the polytropic index, \(\beta _{0}\) is the solidity coefficient of the ground, \(\varepsilon _{0}q(x)\) is the small normal load.
Here, \[ K\left( \omega^{^{\prime }},M,\kappa \right) =1-\left[ 1+\frac{\kappa -1}{2}M\omega ^{\prime }\right] ^{\frac{2\kappa }{\kappa -1}} \] in a one-way flow, \[ K\left( \omega^{^{\prime }},M,\kappa \right) =\left[ 1-\frac{\kappa -1}{2}M\omega^{\prime }\right] ^{\frac{2\kappa }{ \kappa -1}}-\left[ 1+\frac{\kappa -1}{2}M\omega^{\prime }\right] ^{ \frac{2\kappa }{\kappa -1}} \; \] in a two-ways flow, with the boundary condition \[ \omega ^{\prime \prime }(0)=\omega ^{(3)}(0), \omega (1)=\omega^{\prime }(1)=0. \tag{2} \] Similarly, the boundary conditions \[ \omega (0)=\omega^{\prime }(0)=0,\;\omega^{\prime \prime }(1)=\omega^{(3)}(1)=0 \tag{3} \] are studied.
Equation (1) with the boundary condition (2) or (3) is a bifurcation problem (according to the Mach number \(M=M_{0}+\varepsilon \) and the small normal load \(\varepsilon _{0}q\)). A method from I. S. Iohvidov’s work is applied to this problem [Hankel and Toeplitz matrices and forms. Algebraic theory. Stuttgart: Birkhäuser (1982; Zbl 0493.15018)].
The linearized problem \[ \chi ^{2}\omega^{(4)}+\sigma \omega^{^{\prime }}+\beta _{0}\omega =0 \] with condition (3) is investigated. Eigenvalues and eigenfunctions are determined. The adjoint problem is constructed, and the question in which of these cases the problem is divergent or not is answered. The Fredholm property of the linearized problem is proved by means of the construction of the corresponding Green function.