On the numerical solution of the biharmonic equation in the plane

The biharmonic equation arises in a variety of problems in applied mathematics, most notably in plane elasticity and in viscous incompressible flow. Integral equation methods are natural candidates for the numerical solution of such problems, since they discritize the boundary alone, are easy to apply in the case of free or moving boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains, regardless of shape. In this paper, we follow the work of Mayo and Greenbaum and make use of the Sherman-Lauricella integral equation which is a Fredholm equation with bounded kernel. We describe a fast algorithm for the evaluation of the integral operators appearing in that equation. When combined with a conjugate gradient like algorithm, we are able to solve the discretized integral equation in an amount of time proportional to N, where N is the number of nodes in the discretization of the boundary.     

Solutions of the Fokker-Planck equation describing the thermalization of neutrons in a heavy gas moderator

The thermalization of neutrons is described by a transport equation with a second order differential operator with respect to the energy. First this equation is transformed to an one-variable Fokker-Planck equation. Next an eigenfunction expansion and a polynomial expansion are used to solve the time-dependent Fokker-Planck equation. The eigenvalues are obtained either by solving a Schrödinger equation or by calculating 2×2 matrix continued fractions. Explicit results for the approach to equilibrium of a pulse of neutrons as well as the stationary distribution for 1/v absorption are presented. It is shown that the theory of the photoelectromotive force in semiconductors also leads to the same problem.     

sine-Gordon型方程的无穷序列新解

通过下列步骤,获得了sine-Gordon型方程的新解.第一步、通过函数变换,把sine-Gordon方程与sinhGordon方程的求解问题转化为两种非线性常微分方程的求解问题.第二步、获得了两种非线性常微分方程与第一种椭圆方程的拟B?cklund变换.第三步、利用第一种椭圆方程的B?cklund变换与新解,构造了sine-Gordon型方程的无穷序列新解.     

On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation

The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay ratet ^–5/4) ast+ to that of the compressible Navier-Stokes equation for the corresponding initial data.     

混合对流的数值模拟

分别采用(-v'2)-f模型、k-ε模型和室内零方程模型对通风空调中典型的气流组织形式混合对流进行数值模拟.研究表明:在通风入口附近存在一个分离涡,对这个分离涡预测的准确程度是整个计算成败的关键因素.将预测结果和实验进行比较,发现这个涡的尺度介于(-v'2)-f模型和零方程模型预测的结果之间,比k-ε模型预测的结果大.虽然从流场上看,零方程模型预测的结果和实验结果吻合很好,但从模型构造的角度看,它不具备普遍的意义.从对温度场的预测看,(-v'2)-f模型预测结果最好,室内零方程模型次之,k-ε模型结果最差.     

A semi-empirical equation for oxygen nonstoichiometry of perovskite-type ceramics

Explicit equations correlating oxygen nonstoichiometry to oxygen partial pressure and temperature are important for applications of perovskite-type ceramics as membranes, adsorbents and catalysts in various chemical reaction and separation processes. A semi-empirical equation for oxygen nonstoichiometry on perovskite-type ceramics is reported in this paper. Though derived from the results of a point defect model on a perovskite-type ceramic material, La_0.1Sr_0.9Co_0.5Fe_0.5O_3−δ, this equation describes very well the experimentally measured oxygen nonstoichiometry data for two perovskite-type ceramics measured in this work and three perovskite-type ceramics reported in the literature. The major advantage of this semi-empirical equation lies in its simplicity, explicitness and accuracy. This equation is coupled with oxygen permeation equation to predict oxygen permeation current density through two perovskite-type ceramic membranes. The predicted data agree very well with the results reported in the literature using a complex defect reaction model.     

考虑二次梯度项影响的双渗模型的动态特征

在传统试井模型的非线性偏微分方程中根据弱可压缩液体的假设忽略了二次梯度项,已经知道在试井较长时间忽略二次梯度项将产生误差,故对于双重介质流动系统保留了非线性偏微分方程中的二次梯度项,建立了双渗流动模型.采用Douglas-Jones预估校正法获得了无限大地层定产量生产时和定压生产时双渗模型的数值解,分别讨论了液体压缩系数和双重介质参数变化时压力变化规律,做出了典型压力曲线图版,这些结果可用于实际试井分析.     

振动驱动颗粒气体体系的局域态本构关系的实验验证

对准二维、水平边界振动驱动的颗粒气体体系的流体力学 参量进行了局域态本构关系的实验研究. 实验观测结果与经典动力学理论预测进行了比较.由于颗粒气体空间分布的不均匀性, 颗粒体系的整体本构关系不成立, 有必要对局域态进行分析. 局域态本构关系是指颗粒系统的局域温度、局域压强和局域数密度之间的关系. 通过颗粒速度的方向变化, 可以得到颗粒的碰撞点. 因此在计算压力张量的对角线项时, 除了动力学部分之外, 我们计入了颗粒碰撞的影响, 得到了一个约为常数的压力张量迹, 即颗粒压强的空间分布, 与流体力学理论预测以及分子动力学模拟结果相符合; 但是颗粒温度和数密度的空间分布, 在振动的正反两个方向的分量出现差异, 并且温度、压强和数密度之间的局域本构关系, 无论在低密度或高密度区域, 实验与理论预测在定性上一致, 但定量上都有较大差别. 因此经典流体力学理论在描述这样的体系时需加以修正. 关键词： 颗粒气体 态方程 流体力学     

Equation for the effective potential of an atom with the oscillation correction

The equation for the effective potential of an atom in the Thomas-Fermi model is considered. An equation is obtained for the oscillation correction to the potential, due to the nonanalytic dependence of the electron density on Planck's constant . The equation for the exchange and quantum corrections to the potential within the framework of the approximations discussed agrees with the equation of Kompaneets and Pavlovskii. The exchange, quantum, and oscillation corrections to the radial density of the atomic electrons are compared.The author wishes to thank L. Ya. Kobelev for his constant help and encouragement.     

Embedding of Particle Waves in a Schwarzschild Metric Background

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a particle wave, which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the motion equation of a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrödinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrödinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity.     

Activated tunneling decay of metastable state: Solution of the Kramers problem

At sufficiently low friction the Fokker-Planck equation for Brownian motion in a potential well is reduced to an integral equation for the energy variable. The basic small parameter of the problem is the ratio of the temperature T to the depth U₀ of the well. Quantum tunneling effects are naturally incorporated into the calculations. An explicit solution for Kramers' problem of the lifetime of the Brownian particle in the potential well is given.     

LS解法和Fisher方程行波系统的定性分析

提出了求解非线性发展方程的新方法——LS解法.LS解法是基于(G’/G)展开法和扩展的双曲正切函数展开法.并引入了Poincar定性理论的思想,然后以Fisher方程为例进行了试验.通过定性分析首先获得了Fisher方程行波系统积分曲线的性质,然后解得了Fisher方程作为耗散系统时单调减少的波前解和作为扩张系统时单调递增的波前解.一些试验结果与Ablowitz所得结果一致.也得到了Fisher方程作为扩张系统时的新结果.LS解法是在定性理论指导下,在已获知解曲线性质的情况下进行精确求解的,求解目标明确.LS解法揭示了线性系统也可以用作辅助方程来求解非线性系统.     

应用格子Boltzmann模型模拟一类二维偏微分方程

针对一类含源的二维非线性偏微分方程, 通过Chapman-Enskog展开技术和多尺度分析提出了带修正项的简单格子Boltzmann模型. 用模型模拟了几类二维偏微分方程, 数值模拟结果与精确解相符合. 成功将格子Boltzmann方法应用到二维偏微分方程的数值求解中. 关键词： 二维非线性偏微分方程 格子Boltzmann模型 Chapman-Enskog多尺度展开     

The validity of nonlinear Langevin equations

In the presence of internal noise the variables describing a system are intrinsically stochastic. If they constitute a Markov process the expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations. In the lowest or linear noise approximation the fluctuations can be represented by a Langevin equation, provided it is handled appropriately. Higher orders cannot be described by any white noise Langevin equation. The question whether the equation has to be interpreted according to Itô or Stratonovich concerns these higher orders, for which the equation is not valid anyway.     

Field theory of a traveling wave tube amplifier with a tape helix

A self-consistent relativistic field theory for a helix traveling wave tube (TWT) is presented for a configuration in which a magnetized pencil beam propagates through a tape helix enclosed with a loss-free well. A linear analysis of the interaction is solved subject to the boundary conditions imposed by the beam, helix, and wall. The wave equation for the fields within the electron beam corresponds to the Appleton-Hartree magnetoionic wave modes that are of mixed electrostatic/electromagnetic polarization. Hence, the determinantal dispersion equation that is obtained implicitly includes beam space-charge effects without recourse to a heuristic model of the space-charge field. This dispersion equation includes azimuthal variations and all spatial harmonics of the tape helix. Solutions that correspond to both the extraordinary (X) and ordinary (O ) solutions for the Appleton-Hartree modes are found numerically     

Calculation of B/A for n-alkane liquids using the Tait equation

The B/A parameter of acoustic nonlinearity was calculated for a series of n-alkane liquids using the Tait PVT equation of state supplemented with specific heat data. The calculations of sound speed, sound speed derivatives, the two components of B/A, and the value of B/A itself were compared with experimental data taken from the literature and with earlier calculations using a different equation of state. In addition, a comparison of the results with Ballou's rule (linear relation of B/A and reciprocal sound speed) was made. It is concluded that B/A can be calculated from the Tait equation of state with about the same accuracy as direct measurements of sound speed versus pressure and temperature, though the the temperature derivatives of the sound speed are calculated with much lower accuracy than pressure derivatives. The calculations made using the Tait equation are about the same accuracy as calculations made using our equation of state. Also, Ballou's rule does not hold for these liquids.     

(2+1)维广义Calogero-Bogoyavlenskii-Schiff方程的无穷序列类孤子解

为了构造高维非线性发展方程的无穷序列类孤子新解, 研究了二阶常系数齐次线性常微分方程, 获得了新结论. 步骤一, 给出一种函数变换把二阶常系数齐次线性常微分方程的求解问题转化为一元二次方 程和Riccati方程的求解问题. 在此基础上, 利用Riccati方程解的非线性叠加公式, 获得了二阶常系数齐次线性常微分方程的无穷序列新解. 步骤二, 利用以上得到的结论与符号计算系统Mathematica, 构造了(2+1)维广义Calogero-Bogoyavlenskii-Schiff (GCBS)方程的无穷序列类孤子新解. 关键词： 常微分方程 非线性叠加公式 高维非线性发展方程 无穷序列类孤子新解     

THE IMPROVED MURNAGHAN EQUATION

The improved Murnaghan equation is derived by integrating B(P)=-v(∂p/∂v)_T=B₀⁽¹⁾(P)·P and expanding B⁽¹⁾(P) and simllar function in series of (1-K) , where K is the compression ratio (K≡(v/v₀)^1/3. The new equation obtained is compared with Murnagnan equation, Keane equation and Birch equation.It is found that, the improved Murnaghan equation has better convergency in comparison with the Birch equation. The characteristics of the new equation and the reasons of its better convergency are discussed.     

Nonequilibrium statistical mechanics of finite classical systems-II

The work of the previous paper is applied to the study of weakly interacting systems. Either by quasilinear techniques or by analyzing the perturbation series for the smoothed probability density, it is possible to derive a master equation equivalent to that of Brout and Prigogine without requiring the size of the system to become infinite. The properties of this equation are discussed. The equation is self-consistent provided the interactions are weak enough; however, examination of higher terms in the perturbation series shows that their effect might make the master equation invalid for times longer than that taken by a typical particle to cross the containing vessel. In many physical cases, the relaxation time will be shorter than this; also, further studies may show the higher terms to be less important than they seem.Formerly at Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England.