轴向运动三参数黏弹性梁弱受迫振动的渐近分析

研究了轴向运动三参数黏弹性梁的弱受迫振动．建立了轴向运动三参数黏弹性梁受迫振动的控制方程．使用多尺度法渐近分析了运动梁的稳态响应,导出了解稳定性边界方程、稳态振幅的表达式以及稳态响应非零解的存在条件．依据Routh-Hurwitz定律决定了非线性稳态响应非零解的稳定性．     

关于非线性边值问题几个存在性定理的新结果

本文运用同胚理论,研究微分方程边值问题解的存在性与唯一性,得到两个基本定理,推广了Ｂｒｏｗｎ（ｉｎＡｎｎａｌｉ．Ｍａｔ．Ｐｕｒａ．Ａｐｐｌｉ．１９７５,１０６：２０５～２１４）的结论,取消了有界性假设,并把我们的结果用于有限维的情况,考虑了非线性守恒系统在扰动情况下（Ｎｅｗｔｏｎ类运动方程）周期解的存在性与唯一性问题·把同胚用于这类问题的研究目前还是新的     

水动力-热动力学的极值定律

本文对水动力学和更普通性的连续介体动力学中以连续方程与运动方程所表达的现有诸经典守恒定律以外,提出另一最大能量消散率定律.这一定律的推论就是应用水力学中培纶格-波丝最小储存能学说. 凡在运动中消散了的机械能皆转化成为热能,储存在物体里.能量之消散当一定时刻一定温度都使产熵增加.所以,从最大能量消散率可引出热力学第二定律的一个新概念,即机械运动的产熵率也总是一个可能的最大值. 文中建议的这个连续介体极值定律,可从变分原理推导出来,重订热力学第二定律则可藉微观分析加以证明.两者合成水动力-热动力学极值定律     

一类超线性收敛的广义拟Newton算法

１引言考虑无约束最优化问题其中目标函数ｆ（ｘ）二阶连续可微,记ｆｋ＝ｆ（ｘ）,当充分小时,有如下近似关系：它们对二次函数皆严格成立．考虑选代其中Ｂ（Ｇ的近似）已知,为某种线搜索确定的步长．对Ｂ修正产生Ｂ,即Ｕ为待定ｎ阶矩阵．若要求Ｂ＋满足关系即Ｂ满足拟Ｎｅｗｔｏｎ方程,由它可导出许多著名的拟Ｎｅｗｔｏｎ算法［１－［４］）．若要求Ｂ满足关系则可导出伪Ｎｅｗｔｏｎ－δ族校正公式,它不再是Ｈｕａｎｇ族成员［６］．从信息资源的利用看,（１．６）仅利用了与信息,（１．７）仅利用了与信息．一般而言,较多的信…     

大型线性方程组的变分迭代解法

本文提出了一种求解大型线性方程组的一种新方法——变分迭代解法．这种方法的基本思想是：先给方程一个近似的初值,然后引进若干个拉氏乘子校正其近似值,而拉氏乘子可用极值的概念最佳确定．这种方法收敛速度较快,如果只取ｎ个拉氏乘子（ｎ为方程个数）,则该方法即为Ｎｅｗｔｏｎ迭代法．     

用独立变量表示的约束Birkhoff系统的运动稳定性

首先提出Pfaff－Birkhoff－D'Alembert原理，并由此原理导出约束Birkhoff系统用独立交量表示的运动方程；其次建立系统的受扰运动微分方程；最后利用直接法和一次近似理论得到系统运动稳定性的一些判据．     

Newton法及其各种变形收敛性的统一判定法则

１引言Ｂａｎａｃｈ空间中的算子方程是非常广泛的应用数学课题,而求方程数值解的主要方法是Ｎｅｗｔｏｎ法及其各种变形．自从Ｋａｎｔｏｒｏｖｉｃｈ关于Ｎｅｗｔｏｎ法收敛性的著名定理建立以来,有大量的文献在种种相仿的条件下研究各种变形收敛性的各种判定法则．在此,本文建立了统一的判定法则．首先,把Ｋａｎｔｏｒｏｖｉｃｈ算子类Ｋ（１）（ｘ０,ｙ）扩充为Ｋ（１）（ｘ０,ｙ,Ｃ）,并给出类Ｋ（１）（ｘ０,ｙ,Ｃ）的扩类Ｋｃｅｎｔ（１）（ｘ０,ｙ,Ｃ）和子类Ｋ（２）（ｘ０,ｙ,Ｃ）．对于这些算子类,确定了一个仅依…     

矿区土壤中含水率及重金属迁移转化的数值模拟

本文基于由连续性方程和达西定律所推出的土壤中水分运动基本方程,以一维垂向水分方程为研究对象,构造稳定收敛的有限差分格式,运用MATLAB数学工具,对地面饱水情况下土壤水分运动的一维垂向方程进行了数值模拟,得到了土壤中水分的迁移规律;同时,综合考虑对流扩散作用以及土壤对重金属的吸附解吸作用,利用非饱和土壤中重金属离子迁移转化模型,对锌离子在矿区土壤中的迁移转化进行了数值模拟,展示了锌离子在矿区土壤中的浓度分布规律．     

具有快速振荡外力的拟地转运动的平均原理

一类大尺度的地球物理流体流可以用拟地转方程来描述．有限、但是大时间区间和整个时间轴上在快速振荡外力下的拟地转运动的平均原理被得到了．其中包括比较估计,稳定性估计和拟地转运动及其平均运动之间的收敛性．进一步,几乎周期拟地转运动的存在性和吸引子的收敛性也被得到了．     

重建极性连续统理论的基本定律和原理(Ⅹ)——主均衡定律

通过对诸主均衡定律和应用Noether定理得出的守恒定律进行比较,自然地导出微极连续统力学的1个统一的主均衡定律和6个物理上可能的均衡方程．其中,通过扩展众所周知和惯用的能量动量张量的概念,得到相当一般的定名为能量-动量的、能量-角动量的和能量-能量的守恒定律和均衡方程．显然,在这后3种情况下的主均衡定律中,物理场量是难以凭借直觉假定出来的．最后,作为特殊情形,直接推演出若干现有的结果．     

Convergence of a continuous BGK model for conservation laws on the quarter plane

We consider a scalar conservation law in the quarter plan. This equation is approximated in a kinetic BGK model with infinite set of velocities. The convergence is established in the general BV framework, without special restrictions on the flux nor on the equilibrium problem's data.     

Kinetic decomposition for kinetic models of BGK type

In this paper, we consider kinetic models of BGK type which describe the scalar conservation law at the microscopic scale. We use new technique developed in Comm. Partial Differential Equations 27 (2002) 1229 in order to get the convergence. First, we obtain the approximate transport equation for the given kinetic models of BGK type. Then using the averaging lemma, we obtain the convergence. This paper shows how to relate the given kinetic model with the averaging lemma to get the convergence.     

Scaling limits for the Ruijgrok–Wu model of the Boltzmann equation

The Ruijgrok–Wu model of the kinetic theory of rarefied gases is investigated both in the fluid‐dynamic and hydro‐dynamic scalings. It is shown that the first limit equation is a first order quasilinear conservation law, whereas the limit equation in the diffusive scaling is the viscous Burgers equation. The main difficulties came from initial layers that we handle here. Copyright © 2001 John Wiley & Sons, Ltd.     

The second law of thermodynamics for high-energy channeled particles

In microscopic theory, the number of kinetic equations underlying the proof of the second law of thermodynamics is quite restricted. We explicitly prove that the second law of thermodynamics is satisfied for high-energy particles moving in a crystal in the channeling regime. The proof involves a local Boltzmann equation for the distribution function of the particles written in the Bogoliubov form. In this, we take one statistical mechanism into account: the scattering of channeled particles on lattice atoms randomly displaced from the crystal sites.     

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.     

Fractional diffusion limit for a stochastic kinetic equation

We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.     

Long-time behaviour of a one-dimensional BGK model: convergence to macroscopic rarefaction waves

For one-dimensional kinetic BGK models, regarded as relaxation models for scalar conservation laws with genuinely nonlinear fluxes, we prove that the macroscopic density converges to the rarefaction wave solution of the corresponding scalar conservation law in the long time limit, and that the phase space density approaches an equilibrium distribution with the rarefaction wave as macroscopic density. The proof requires a smallness assumption on the amplitude of the rarefaction waves and uses a micro-macro decomposition of the perturbation equation.     

Long-time behaviour of a one-dimensional BGK model: convergence to macroscopic rarefaction waves

For one-dimensional kinetic BGK models, regarded as relaxation models for scalar conservation laws with genuinely nonlinear fluxes, we prove that the macroscopic density converges to the rarefaction wave solution of the corresponding scalar conservation law in the long time limit, and that the phase space density approaches an equilibrium distribution with the rarefaction wave as macroscopic density. The proof requires a smallness assumption on the amplitude of the rarefaction waves and uses a micro-macro decomposition of the perturbation equation.     

Uncertainty principle and kinetic equations

A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation.