非线性抛物型时滞微分方程解振动的充要条件

讨论了一类多滞量非线性抛物型时滞微分方程解的振动性质，获得了其一切解振动的充要条件；指出了其与普通抛物型偏微分方程解的质的差异。     

非线性中立双曲型时滞微分方程解的振动判据

讨论一类多滞量非线性中立双曲型时滞微分方程的振动性质，获得了其一切解振动的充分条件。     

非线性中立抛物型时滞偏微分方程解的振动性质

讨论一类多滞量非线性中立抛物型时滞偏微分方程解的振动性质 ,应用积分不等式和泛函微分方程的某些结果 ,在第一类边界条件下获得了其一切解振动的一系列充分条件 .结论充分表明了时滞量的决定性作用 ,指出了其与普通抛物型偏微分方程质的差异 .     

高阶非线性阻尼脉冲微分方程解的振动性

研究了一类高阶非线性阻尼脉冲微分方程振动状态,得到振动解的充分条件.     

非线性中立抛物型泛函微分方程解的振动判据

讨论一类多滞量中立抛物型泛函微分方程解的振动性质,获得了其一切解振动的充分条件及线性情况下的充要条件;指出了与普通抛物型偏微分方程质的差异。     

非线性中立抛物型时滞偏微分方程解的振动性质

讨论一类多滞量非线性中立抛物型时滞偏微分方程解的振动性质 ,获得了其一切解振动的充分条件 ;指出了与普通抛物型偏微分方程质的差异 .     

非线性中立双曲型时滞偏微分方程解的振动性质

讨论一类多滞量非线性中立双曲型时滞偏微分方程解的振动性质,应用积分不等式和泛函微分方程的某些结果,获得了其一切解振动的一系列充分条件。结论充分表明了时滞量的决定性作用,指出了其与普通双曲型偏微分方程质的差异。     

中立抛物型偏微分方程解的强迫振动

本文讨论一类多滞量中立抛物型偏微分方程解的振动性质 ,获得了其一切解振动的充分条件 ;指出了与普通抛物型偏微分方程质的差异 .     

非线性中立双曲型泛函微分方程解的振动判据

讨论一类多滞量非线性中立双曲型泛函微分方程解的振动性质，获得了其一切解振动的充分条件；指出了与普通双曲型偏微分方程质的差异。     

中立双曲型时滞偏微分方程解振动的充要条件

本文讨论一类线性中立双曲型时滞微分方程解的振动性质，获得了其一切解振动的充要条件．     

Oscillations of a Prismatic Rod

Equations are obtained for the longitudinal and transverse oscillations of a prismatic rod that differ from the standard equations of structural mechanics (strength of materials). The difference lies both in the appearance of new terms in the equations of motion corresponding to the Timoshenko theory, and in a refinement of the boundary conditions and the conditions on the rigidity of the rod for transverse oscillations. The oscillations of a vertical cantilevered rod when its base moves horizontally from a state of rest are examined and a comparison is made with the classical theory of transverse (bending) oscillations of prismatic rods.     

NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS OF PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH DELAY

In this paper, by making use of the integral method and some results of the functional differential equations, oscillatory properties of the solutions of certain parabolic partial differential equations with multi-delays are investigated and a series of necessary and sufficient conditions for oscillations of the equations are established. The results fully indicate that the oscillations are caused by the delay and hence reveal the varied difference between these equations and those without delay.     

The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.     

中立抛物型时滞偏微分方程解振动的充要条件

本丈讨论一类多滞量中立抛物型时滞偏微分方程解的振动性质。获得了其一切解振动的充要条件；所得充要条件将时滞偏微分方程解的振动判别问题转化为时滞微分方程解的振动判别问题；指出了其与普通抛物型偏微分方程解的质的差异．     

Numerical Study of Oscillatory Solutions of the Gas-Dynamic Equations

Solutions of the one-dimensional gas-dynamic equations with oscillatory initial data are studied numerically. A fourth-order ENO scheme is used, together with an artificial compression method to effectively eliminate the smearing of contact discontinuities. The numerical results show that the oscillations die out in velocity and pressure, and persist in density. A system of equations governing the propagation of these oscillations is derived.     

Numerical Stability and Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Advanced Type

For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.     

An asymptotic approach to the problem of the free oscillations of a beam

Equations describing the free small longitudinal and transverse oscillations of a straight elastic beam of rectangular cross section are obtained using the plane linear theory of elasticity and the method of integrodifferential relations. The initial system of partial differential equations is reduced to a system of ordinary linear differential equations with constant coefficients. The effect of the geometrical and elastic characteristics of the beam on the frequency and form of the natural oscillations is investigated. For longitudinal motions it is shown that different types of natural displacements and internal stresses of the beam exist. For transverse oscillations, it is found that there are frequency zones corresponding to different forms of the solutions of the characteristic equation obtained using the proposed model.     

Non-linear radial oscillations of a transversely isotropic hyperelastic incompressible tube

The constitutive equation for a transversely isotropic incompressible hyperelastic material is written in a covariant form for arbitrary orientation of the anisotropic director. Three non-linear differential equations are derived for radial oscillations in radial, tangential and longitudinal transversely isotropic thin-walled cylindrical tubes of generalised Mooney-Rivlin material. A Lie point symmetry analysis is performed. The conditions on the strain-energy function and on the net applied surface pressure for Lie point symmetries to exist are determined. For radial and tangential transversely isotropic tubes the differential equations are reduced to Abel equations of the second kind. Radial oscillations in a longitudinal transversely isotropic tube and in an isotropic tube are described by the Ermakov-Pinney equation.     

抛物型时滞偏微分方程解振动的充要条件

讨论一类多滞量抛物型时滞偏微分方程解的振动性质，获得了其一切解振动的充要条件及一些充分条件；指出了其与普通抛物型偏微分方程解的质的差异．     

OSCILLATIONS OF FOURTH ORDER ODE WITH IMPULSES

Sufficient conditions to guarantee the oscillations of fourth order ODE with impulses are obtained. The importance of the impulsive effect on the oscillations of higher order differential equations is stressed.