Construction of the asymptotics of solutions of a nonlinear boundary value problem for a fourth order differential equation with two bifurcation parameters (Q447653)

An integro-differential equation of fourth order is considered which describes the deflection of a plate in supersonic gas flow: NEWLINE\[NEWLINE\chi ^{2}\left( \frac{\omega ^{\prime \prime }}{\left( 1+\omega ^{\prime ^{2}}\right) ^{\frac{3}{2}}}\right) ^{\prime \prime }+\beta _{0}\omega =kK\left( \omega^{^{\prime }},M,\kappa \right)+\theta \omega ^{\prime \prime }\int_{0}^{1}\left[ \left( 1+\omega ^{\prime ^{2}}\right) ^{\frac{1}{2}}-1\right] dx+\varepsilon _{0}q(x),\tag{1}NEWLINE\]NEWLINE 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.NEWLINENEWLINEHere, NEWLINE\[NEWLINE K\left( \omega^{^{\prime }},M,\kappa \right) =1-\left[ 1+\frac{\kappa -1}{2}M\omega ^{\prime }\right] ^{\frac{2\kappa }{\kappa -1}} NEWLINE\]NEWLINE in a one-way flow, NEWLINE\[NEWLINE K\left( \omega^{^{\prime }},M,\kappa \right) =\left[ 1-\frac{\kappa -1}{2}M\omega^{\prime }\right] ^{\frac{2\kappa }{ \kappa -1}}-\left[ 1+\frac{\kappa -1}{2}M\omega^{\prime }\right] ^{ \frac{2\kappa }{\kappa -1}} \;NEWLINE\]NEWLINE in a two-ways flow, with the boundary condition NEWLINE\[NEWLINE \omega ^{\prime \prime }(0)=\omega ^{(3)}(0), \omega (1)=\omega^{\prime }(1)=0. \tag{2}NEWLINE\]NEWLINE Similarly, the boundary conditions NEWLINE\[NEWLINE \omega (0)=\omega^{\prime }(0)=0,\;\omega^{\prime \prime }(1)=\omega^{(3)}(1)=0 \tag{3}NEWLINE\]NEWLINE are studied.NEWLINENEWLINEEquation (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 \textit{I. S. Iohvidov}'s work is applied to this problem [Hankel and Toeplitz matrices and forms. Algebraic theory. Stuttgart: Birkhäuser (1982; Zbl 0493.15018)].NEWLINENEWLINEThe linearized problem NEWLINE\[NEWLINE \chi ^{2}\omega^{(4)}+\sigma \omega^{^{\prime }}+\beta _{0}\omega =0 NEWLINE\]NEWLINE 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.