解析式状态方程计算含临界点方阱流体的性质

含临界区流体的物性计算一直是热力学领域的难点。一般认为，解析式状态方程不可能同时很好地计算远离临界点和靠近临界点流体的热力学性质。该文目的就是验证解析式方程在计算热力学性质特别是接近临界点附近的热力学性质适用性。通过计算机分子模拟数据，研究了各种解析式状态方程计算接近临界区的方阱流体的物性的可能性，结果表明，经过参数重新拟合新的解析式方程，可以很好地计算方阱流体的相平衡、密度等性质，直到对比温度接近临界点0．05处。临界点温度、压力和密度的计算值和分子模拟值也基本符合。该文结果可以为解析式方程计算范围的扩大提供依据，从而试图解析式在工程方面有更加广泛的应用。     

Linearization methods in classical and quantum mechanics

The applicability and accuracy of linearization methods for initial-value problems in ordinary differential equations are verified on examples that include the nonlinear Duffing equation, the Lane-Emden equation, and scattering length calculations. Linearization methods provide piecewise linear ordinary differential equations which can be easily integrated, and provide accurate answers even for hypersingular potentials, for which perturbation methods diverge. It is shown that the accuracy of linearization methods can be substantially improved by employing variable steps which adjust themselves to the solution.     

Finite element method analysis of Fokker–Plank equation in stationary and evolutionary versions

The problems that often arise in stochastic dynamics can be investigated using the Fokker–Planck (FP) equation. The response of a such systems being subjected to additive and/or multiplicative random noise is represented by probability density function (PDF) that gives the full information about a response random character. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful alternative. In particular the Finite Element Method (FEM) seems to be very effective. A couple of single dynamic linear/non-linear (Duffing and Van Der Pol type) systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation. The resulting PDFs are analyzed and if the analytic results exist mutually compared.     

Stability of θ-schemes for partial differential equations with piecewise constant arguments of alternately retarded and advanced type

ABSTRACT

This paper deals with the analytical and numerical stability of a partial differential equation with piecewise constant arguments of alternately retarded and advanced type. Firstly, the theory of separation of variables in matrix form and the Fourier method are implemented to achieve the sufficient conditions under which the analytic solution is asymptotically stable. Secondly, the discrete equation is obtained by applying the θ-schemes to the original continuous equation, the sufficient conditions for the asymptotic stability of numerical solution are also shown when the mesh ratio satisfying certain conditions. Finally, some numerical experiments for verifying the theoretical results are provided.     

H∞状态反馈控制及其最优解的探讨

本文研究了状态完全可得时Ｈ＾∞状态反馈控制问题，给出了基于一个代数Ｒｉｃｃａｔｉ方程的Ｈ＾∞状态反馈控制问题的次优解，并对一种特殊情况提出了求最优解的解析方法。     

深亚微米器件玻耳兹曼传输方程的实用解析模型

深亚微米MOSFET须应用玻耳兹曼传输模型。本文建立了一个基于物理假定的解析解模型，其特点是可以加入经典模型BSIMGV3，并通过模拟程序CMOSIS进行了模拟验证。     

Piecewise quasilinearization techniques for singular boundary-value problems

Piecewise quasilinearization methods for singular boundary-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. The accuracy of the globally smooth piecewise quasilinear method is assessed by comparisons with exact solutions of several Lane-Emden equations, a singular problem of non-Newtonian fluid dynamics and the Thomas-Fermi equation. It is shown that the smooth piecewise quasilinearization method provides accurate solutions even near the singularity and is more precise than (iterative) second-order accurate finite difference discretizations. It is also shown that the accuracy of the smooth piecewise quasilinear method depends on the kind of singularity, nonlinearity and inhomogeneities of singular ordinary differential equations. For the Thomas-Fermi equation, it is shown that the piecewise quasilinearization method that provides globally smooth solutions is more accurate than that which only insures global continuity, and more accurate than global quasilinearization techniques which do not employ local linearization.     

一维位置敏感探测器的计算机仿真

位置敏感探测器(PSD)是基于PN结横向光电效应的光电器件，其数学模型为Lucovsky方程，通常在二维区域，该方程只在特定条件下可得到解析解。一维PSD具有一定的物理宽度，该宽度对PSD的影响由于受到边界条件限制很难用解析法分析。该文采用偏微分方程求解软件FlexPDE对一维PSD在二维区域内进行了数值仿真，结果表明，宽度较窄的一维PSD其定位误差较宽度大的一维PSD要小，在PSD给定的情况下，光斑在PSD宽度方向上的移动对定位误差基本没有影响。     

High accuracy spline interpolation for 5-axis machining

This article presents a new algorithm for simultaneous 5-axis spline interpolation. The algorithm basically merges two concepts: (1) the interpolation of the toolpath with Pythagorean Hodograph (PH) curves and (2) the analytic solution of the inverse kinematic problem using the template equation method. The first method allows one to obtain a analytic relation between the arclength and the parameter of the toolpath curve. This one enables one to control the velocity of the tool on the workpiece. The second method allows one to determine the analytic solution of the parameterized inverse kinematic problem that permits us to introduce arbitrary number of geometric parameters. A natural selection of the possible parameters can be the parameters of tool geometry and the workpiece placement. This way, the off-line generated inverse solutions—that transform the cutter contact curve into axis values—can be online compensated, as soon as the exact parameter values are become known. Based on these two approaches, a robust and fast method for the simultaneous 5-axis spline interpolation is developed. The result of this new algorithm is time-dependent axis splines which represent the given toolpath with high accuracy.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

地球J_2摄动下卫星编队飞行自主相对定轨研究

卫星编队飞行自主相对定轨作为实现编队队形保持和控制的前提,是编队任务必须解决的关键技术之一.设计了一种应用于卫星编队的自主相对轨道确定方案,不同于目前广泛采用的C-W方程,采用了包含摄动影响的相对运动方程的解析表达式描述编队飞行,使得模型的精度有了很大提高.利用星间距离信息和方位信息作为观测量,设计扩展卡尔曼滤波器实现环绕星相对轨道的自主确定,仿真结果表明使用本方案得到的定轨精度比采用C-W方程提高一个数量级,验证了这种导航方案的有效性,为实现编队飞行卫星高精度自主相对定轨提供了一个有效的方案.     

线性二次最优控制的精细积分法

LQ控制虽然是最优控制的最基本问题,但其数值求解仍有很多问题.黎卡提微分 方程的精细积分法利用黎卡提方程的解析特点,求出计算机上高度精密的解,并已证明误差 在计算机倍精度数的误差范围之外.这对于Kalman-Bucy滤波,LQG问题以及H∞控制及滤 波等都可运用,精细积分还求解了反馈后的状态微分方程.数例验证了其高精度特性.     

A method for solving nth order fuzzy linear differential equations

In this paper, an analytic method (eigenvalue–eigenvector method) for solving nth order fuzzy differential equations with fuzzy initial conditions is considered. In this method, three cases are introduced, in each case, it is shown that the solution of differential equation is a fuzzy number. In addition, the method is illustrated by solving several numerical examples.     

编队飞行卫星高精度自主相对定轨研究

设计了一种应用于卫星编队的自主相对轨道确定方案,不同于目前广泛采用的C-W方程,采用了包含摄动影响的相对运动方程的解析表达式描述编队飞行。利用星间距离信息和方位信息作为观测量,设计扩展卡尔曼滤波器实现环绕卫星相对轨道的自主确定,仿真结果表明定轨精度比采用C-W方程提高一个数量级,验证了这种导航方案的有效性。     

求解二维对流扩散方程的格子Boltzmann方法

针对二维对流扩散方程,基于D2Q4格子速度,用Chapman-Enskog多尺度分析技术,将时间尺度取为二阶,空间尺度取为一阶,推导了各个速度方向上的平衡态分布函数所满足的条件,给出了简单且对称的平衡态分布函数表达式,所得到的平衡态分布函数能正确地恢复出二维对流扩散方程,从而构建了一种新的求解二维对流扩散方程的D2Q4格子Boltzmann（LB）模型。用所给LB模型对扩散方程和两个不同初边界条件的对流扩散方程进行了数值求解,数值实验结果表明数值解与精确解吻合较好,与相关文献结果比较边界误差要小得多,验证了模型的有效性。     

An analytic approach for computing pi-coefficients of continuous-time systems

The pi-sharing theory for continuous-time systems is discussed in this paper. An analytic approach is proposed for computing pi-coefficients of LTI systems. For the two scalar pi-coefficients, feasible regions are characterized graphically on a plane, and for the other two matrix pi-coefficients, the corresponding values are obtained by solving a Riccati equation. The results are applied in an example to determine the stability of a feedback system containing both linear and nonlinear subsystems.     

Application of the Boltzmann kinetic equation to the eddy problems

The main purpose of this work is to investigate the feasibility of applying a kinetic approach to the problem of modeling turbulent and unstable flows. First, initial value problems with the Taylor–Green (TG) type and isotropic velocity conditions for compressible flow in two-dimensional (2D) and three-dimensional (3D) periodic domains are considered. Further, 3D direct numerical simulation of decaying isotropic turbulence is performed. Macroscopic flow quantities of interest are examined. The simulation is based on the direct numerical solution of the Boltzmann kinetic equation using an explicit–implicit scheme for the relaxation stage. Comparison with the solution of the Bhatnagar–Gross–Krook (BGK) model equation obtained by using an implicit scheme is carried out for the decaying isotropic turbulence problem and demonstrates a small difference. For the TG initial condition results show a fragmentation of the large initial eddies and subsequently the full damping of the system. Numerical data are close to the analytic solution of TG problem. A dependence of the kinetic energy on the wave number is obtained by means of the Fourier expansion of velocity components. A power-law exponent for the kinetic energy spectrum tends to the theoretical value “−3” for 2D turbulence in 2D case and to the famous Kolmogorov value “−5/3” in 3D case.     

Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method

In this paper, we propose a regular perturbation method to obtain approximate analytic solutions of exterior and interior Dirichlet problems for Laplace’s equation in planar domains. This method, starting from a geometrical perturbation of these planar domains, reduces our problems to a family of classical Dirichlet problems for Laplace’s equation in a circle. Numerical examples are given and comparisons are made with the solutions obtained by other approximation methods.     

Application of the Exp-function method to the (2+1)-dimensional Boiti–Leon–Pempinelli equation using symbolic computation

This paper deals with the so-called Exp-function method for studying a particular nonlinear partial differential equation (PDE): the (2+1)-dimensional Boiti–Leon–Pempinelli equation. The method is constructive and can be carried out in a computer with the aid of a computer algebra system. The obtained generalized solitary wave solutions contain more arbitrary parameters compared with the earlier works, and thus, they are wider. This means that our method is effective and powerful for constructing exact and explicit analytic solutions to nonlinear PDEs.     

Solitary-wave solutions of the Degasperis–Procesi equation by means of the homotopy analysis method

The homotopy analysis method (HAM) is applied to the Degasperis–Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the HAM is a powerful tool for finding excellent approximations to nonlinear solitary waves.