Construction of the asymptotics of solutions of a nonlinear boundary value problem for a fourth order differential equation with two bifurcation parameters
type: Item
instance of: scholarly article
title: Construction of the asymptotics of solutions of a nonlinear boundary value problem for a fourth order differential equation with two bifurcation parameters
zbMATH Open document ID: 1267.34035
author: Tatiana E. Badokina
author: Boris V. Loginov
author: Yuri B. Rousak
publication date: 2012-09-05T00:00:00Z
reviewed by: Kh. R. Mamedov
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34B60
Mathematics Subject Classification ID: 76J99
zbMATH DE Number: 6077045
zbMATH Keywords: boundary value problem of ordinary differential equations of higher order
zbMATH Keywords: multi-parameter bifurcation
zbMATH Keywords: plate in supersonic gas flow
zbMATH Keywords: plate deflection
zbMATH Keywords: discriminant curve
zbMATH Keywords: branching equation
MaRDI profile type: Publication
published in: The Bulletin of Irkutsk State University. Series Mathematics
review text: 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 appli…
description: scientific article; zbMATH DE number 6077045