Solvability of 2n-order m-point boundary value problem at resonance

The existence of solutions for the 2n-order m-point boundary value problem at resonance is obtained by using the coincidence degree theory of Mawhin.We give an example to demonstrate our result.The interest is that the nonlinear term may be noncontinuous.     

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method.Moreover,it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.     

基于区间参数摄动法的三维弹塑性区间有限元探讨

首次探讨了弹塑性问题的区间分析,推导了三维弹塑性问题的区间参数摄动法有限元计算公式,研制了相应的弹塑性区间有限元程序,通过算例分析得到结论:(1)采用增量法对弹塑性问题进行区间分析时,增量位移离差随着增量位移均值的增大而增大.(2)单元高斯点应力没有达到屈服极限之前,高斯点的应力离差很小;当单元高斯点应力达到屈服极限之后,高斯点应力离差逐渐增大.(3)是否考虑塑性矩阵的应力分量为弹塑性参数的隐函数,对应力高差计算的影响较小.     

Solution of two-dimensional scattering problem in piezoelectric/piezomagnetic media using a polarization method

Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezo-magnetic "comparison body" is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green's function, which is ob-tained using the Radon transform. The expression is further simplified under condi-tions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelec-tric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.     

Asymptotic modeling of the impact of a spherical indenter on an elastic half-space

The impact of a rigid sphere on a homogeneous, isotropic elastic half-space in the absence of friction and adhesion is considered. The influence of the superseismic stage immediately following the moment of first contact upon the impact process is investigated in the frame of the Hertzian impact theory. The first order asymptotic approximation for the contact force in a three-dimensional dynamic contact problem with the slowly moving contact zone boundary is obtained and the corresponding asymptotic model of impact is developed. The motion of the indenter as it indents and rebounds from the elastic medium is analytically described. Explicit formulas are derived for the peak indentation depth, contact time, and rebound velocity as functions of the initial impact velocity, indenter mass, and characteristics of the elastic half-space.     

Acoustic stop bands in almost-periodic and weakly randomized stratified media： perturbation analysis

In this paper, we analyze the effect of both deter- ministic and random perturbations of a regular multi-layered elastic structure on its stop band properties. The tool of choice is the transfer matrix method, which is both versatile and easy to implement. In both cases, we find that the stop-bands widen. We observe the appearance of very narrow pass-bands within the stop-bands, which can be observed in other instances in optics.     

Vertical motion of a spherical body with decreasing mass

The paper studies the fall and ascent in air of a spherical body with radius linearly decreasing with time. The first integral of the nonlinear equation of motion is expressed in closed form in terms of Bessel functions on the assumption that the aerodynamic drag depends on the velocity as a quadratic polynomial. The second integral is evaluated by an approximate formula. The reliability of the analytic solution is confirmed by comparing it with a numerical solution of the Cauchy problem __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 118–125, June 2008.     

A two-body continuous-discrete filter

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodeled stochastic acceleration is not assumed in the particle dynamics. The small, unmodeled stochastic acceleration produces an additional random force on a particle. Estimation algorithms for a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. Specifically, this paper examines the effect of the stochastic acceleration on the motion of the orbiting particle, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means and conditional variances for estimating the states of the particle-earth system. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation “between the observations" and a functional difference equation for the conditional probability density “at the observation." The effectiveness of the nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting particle and the signal-to-noise ratio. The Kolmogorov forward equation, however, is not appropriate for the numerical simulations, since it is the equation for the evolution of “the conditional probability density." Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting body. Even these equations are not appropriate for the numerical implementations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher order moments. Hence, we consider the approximations to these moment evolution equations. This paper makes a connection between classical mechanics, statistical mechanics and the theory of the nonlinear stochastic filtering. The results of this paper will be of use to astrophysicists, engineers and applied mathematicians, who are interested in applications of the nonlinear filtering theory to the problems of celestial and satellite mechanics. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed, in this paper.     

An analytical solution to non-axisymmetric heat transfer with viscous dissipation for non-Newtonian fluids in laminar forced flow

Forced convection heat transfer in a non-Newtonian fluid flow inside a pipe whose external surface is subjected to non-axisymmetric heat loads is investigated analytically. Fully developed laminar velocity distributions obtained by a power-law fluid rheology model are used, and viscous dissipation is taken into account. The effect of axial heat conduction is considered negligible. The physical properties are assumed to be constant. We consider that the smooth change in the velocity distribution inside the pipe is piecewise constant. The theoretical analysis of the heat transfer is performed by using an integral transform technique – Vodicka’s method. An important feature of this approach is that it permits an arbitrary distribution of the surrounding medium temperature and an arbitrary velocity distribution of the fluid. This technique is verified by a comparison with the existing results. The effects of the Brinkman number and rheological properties on the distribution of the local Nusselt number are shown.     

Applications of new affine invariant for polytopes

To study the Schneider＇s projection problem, Lutwak, Yang and Zhang recently introduced a new .affine invariant functional U（P） for convex polytopes in R^n. In the paper, we obtain the analytic expression of the affine-invariant U（P） defined on a specific subclass of origin-symmetric convex polytopes in Rn and give an application of U（P） to the Lp-Minkowski problem.