学会方程思想方法

<正>所谓方程思想方法,就是以方程的视角审视问题,通过建立相关的方程来解决问题的思想方法.由于方程思想方法贯穿了高中数学,因此加深对方程思想方法的理解掌握,提高运用方程思想方法解决问题能力,是学好高中数学的重要方面.本文拟从三个方面就如何学会方程思想方法,向同学们提出建议,供参考.一、增强方程意识所谓方程意识,即是运用方程角度看问题的意识.增强方程的意识,就是要养成一种当     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发,推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示,三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到,表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

Rangwala-Rao方程和几个非线性导数Schrdinger型方程的一类显式精确孤波解

首先求出了Lienard方程的显式精确解,进而求出了Rangwala-Rao方程,Ablowitz方程,Chen-Lee-Lin方程,以及Gerdjikov-Ivanov方程的型如的显式精确孤波解。     

一类广义变系数mKdV方程的B\"{a}cklund变换, Painlev\'{e}检验及精确解[英文]

本文研究一类广义变系数mKdV方程, 基于齐次平衡法, 对方程进行B\"{a}cklund变换, 进而得到方程的精确解; 对方程进行Painlev\''{e}检验, 证明方程的可积性. 利用推广的CK方法, 将广义变系数mKdV方程化为常系数方程, 结合幂级数法得到方程的幂级数解.     

应用(1/G)-展开法求解五阶KdV方程

本篇论文首次提出(1/G) -展开法,用于求解非线性演化方程的行波解.将该法应用于五阶KdV方程的求解,当参数满足一定条件时,该方程可化为Sawada-Kotera (SK)方程、Caudrey-Dodd-Gibbon(CDG)方程、Kaup-Kupershmidt (KK)方程、Lax方程和Ito方程.其解可被表示为...     

推广的Boussinesq方程和KdV方程——Painlevé性质,Bcklund变换和Lax对

本文从Kadomtsev-Petviashvili方程的对称性约化出发,得到了具有任意函数作为变系数的推广的Boussinesq方程和推广的KdV方程.利用Weiss和Kruskal等建立的奇性分析方法,我们证明了这两个方程可积的充分条件:Painlevé性质。得到了这两个方程的Bcklund变换和奇性流形方程(推广的Schwartz-Boussinesq方程和Schwartz-KdV方程)。并由此将这两个方程线性化,即给出这两个方程的Lax对并包含有明显的与时间无关的任意谱参数。     

有界区域上的粘性分离法

用分步法求解Navier-Stokes方程的初边值问题。在每一个时间区间内,将原方程分解为没有扩散项的Euler方程和没有对流项的扩散方程,证明了近似解的收敛性。与通常的方法不同的是,我们用一个非齐次扩散方程代替了齐次扩散方程。本文是一系列文章的第一篇,讨论线性化的方程。     

三维重调和方程的双方程边界积分方程法

以守恒积分为工具,推导了三维重调和方程的新的边界积分方程,所得出的新方程与传统的边界积分方程相比较,降低了奇异性,避免了传统边界元方法中的强奇异积分的计算.对不同边界都采用第二类积分方程,得到了三维重调和方程的双方程方法.     

学会方程思想方法

所谓方程思想方法,就是以方程的视角审视问题,通过建立相关的方程来解决问题的思想方法.由于方程思想方法贯穿了高中数学,因此加深对方程思想方法的理解掌握,提高运用方程思想方法解决问题能力,是学好高中数学的重要方面.本文拟从三个方面就如何学会方程思想方法,向同学们提出建议,供参考.     

不可压流体的边界层问题

研究三维有界区域在边界上有流动的不可压流体的边界层问题,导出了Navier-Stokes方程区域内部的近似方程(Euler方程和线性化的Euler方程)和边界附近近似的方程(零阶边界层方程与一阶边界层方程),证明了这种近似的合理性.     

On the algorithm of the solution of the signorini problem

An algorithm is proposed for solving the Signorini problem /1/ in the formulation of a unilateral variational problem for the boundary functional in the zone of possible contact /2/. The algorithm is based on a dual formulation of Lagrange maximin problems for whose solution a decomposition approach is used in the following sense: a Ritz process in the basis functions that satisfy the linear constraint of the problem, the differential equation in the domain, is used in solving the minimum problem (with fixed Lagrange multipliers); the maximum problem is solved by the method of descent (a generalization of the Frank-Wolf method) under convexity constraints on the Lagrange multipliers. The algorithm constructed can be conisidered as a modification of the well-known algorithm to find the Udzawa-Arrow-Hurwitz saddle points /3, 4/. The convergence of the algorithm is investigated. A numerical analysis of the algorithm is performed in the example of a classical contact problem about the insertion of a stamp in an elastic half-plane under approximation of the contact boundary by isoparametric boundary elements. The comparative efficiency of the algorithm is associated with the reduction in the dimensionality of the boundary value problem being solved and the possibility of utilizing the calculation apparatus of the method of boundary elements to realize the solution.     

宏观因素影响下的系统中元件重要性研究

为研究复杂系统在工作环境中其组成元件对系统安全运行的重要性,将汪培庄先生的因素空间理论与笔者提出的空间事故树理论相结合,构造了一套元件重要性研究方法.构建系统T={U,C,D},将元件作为研究对象集合U,系统工作的宏观环境作为因素集C,元件重要性排序集作为D.对宏观环境中的工作时间a1和温度a_2进行划分形成不同的状态区域S_q,计算在S_q中元件xj的失效权重γ(AS_q(x_j))和在S_q中系统T的失效权重δ(AS_q(T))),从而得到x_j在S状态下的等效失效权重Z(AS_q(x_j)),研究状态S_q下的原件重要性排序D_η,及元件x_j失效性对a_1及a_2的敏感性.使用一个实际的电气系统维修情况统计资料,使用上述方法进行了研究,结果表明:不同工作环境下元件对系统的重要程度是不同的.元件对温度和使用时间是敏感的,并得到了在10³0°且5030°且50⁷5d环境下工作系统可靠性是最高的结论.在给定工作环境下,重要性大的元件多储备,重要性小的元件少储备,以满足系统维修需要,并指导实际工程.     

Acceleration of the convergence of the Laguerre series in the problem of inverting the Laplace transform

When the Laplace transform is inverted numerically, the original function is sought in the form of a series in the Laguerre polynomials. To accelerate the convergence of this series, the Euler-Knopp method is used. The techniques for selecting the optimal value of the parameter of the transform on the real axis and in the complex plane are proposed.     

On the optimality of the greedy solutions of the general knapsack problems

Summary: In this paper we submit a unified discussion of some closely related results which were achieved independently in number theory and integer programming, and we partially generalize them. In the unified discussion we treat together two problems where the greedy method has different characters, in the first one it is an internal-point algorithm, in the second one it is an outer-point method. We call a knapsack problem "pleasant" if the greedy solution is optimal for every right-hand side. A sufficient and two finite necessary and sufficient conditions for the pleasantness of a problem are discussed. The sufficient condition can be checked very easily. The paper is finished with an error analysis of some nonpleasant problems     

On the accuracy of the projection computation of the collision integral

The accuracy of the projection method as applied to the computation of the collision integral is analyzed. It is shown that the method has an error of the second order of smallness with respect to the mesh size. An optimal method for choosing additional nodes that minimizes the computational error is found. The theoretical conclusions and the optimality of the method are confirmed in a series of numerical experiments.     

The numerical analysis of the influence of the blankholder force and the friction coefficient on the value of the drawing force

In this paper the numerical analysis of the drawing process with the blankholder is presented. The influence of the value of the blankholder force on the value of the drawing force and drawpiece height is examined. A different friction coefficient of the sheet metal for die and blankholder is considered. A defect at the lack of the blankholder is presented. The analysis in the system ANSYS/LS-DYNA is passed. The material model with the mixed hardening, isotropic and isothermal is used. Technological parameters of the sheet-metal to drawing, the die, the stamp and the blankholder from the literatures are chosen. Examples results of computer simulations are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the minimization of the product of the powers of several integrals

This paper considers the minimization of the product of the powers ofn integrals, each of which depends on a functiony(x) and its derivative . The necessary conditions for the extremum are derived within the frame of the Mayer-Bolza formulation of the calculus of variations, and it is shown that the extremal arc is governed by a second-order differential equation involvingn undetermined multipliers related to the unknown values of the integrals. After the general solution is combined with the definitions of the multipliers and the end conditions, a system ofn+2 algebraic equations is obtained; it involvesn+2 unknowns, that is, then undetermined multipliers and two integration constants.The procedure discussed here can be employed in the study of shapes which are aerodynamically optimum at supersonic, hypersonic, and free-molecular flow velocities, that is, wings and fuselages having the maximum lift-to-drag ratio or the minimum drag. The problem of a slender body of revolution having the minimum pressure drag in Newtonian hypersonic flow is developed as an example. First, a general solution is derived for any pair of conditions imposed on the length, the thickness, the wetted area, and the volume. Then, a particular case is treated, that in which the thickness and the wetted area are given, while the length and the volume are free; the shape minimizing the pressure drag is a cone.This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensed version of the investigation described in Ref. 1. The author is indebted to Messrs. H. Y. Huang, J. C. Heideman, and J. N. Damoulakis for analytical and numerical assistance.     

Reversal of the reinforcing effect of fillers and the relation between the flexibility of the macromolecules and the mechanical properties of filled polymers

The author describes various effects which he has detected involving the reversal of the reinforcing action of fillers in polymers in different subregions of the glassy state. It is suggested that these effects can be explained in terms of, among other things, the flexibility of the macromolecules. The concept of an optimal flexibility at which the maximum value of a given mechanical property can be realized is introduced. It is shown that it is possible, in principle, to reinforce a polymer in any subregion of the glassy state and even below T_b, as well as to obtain equal-strength, equal-modulus, equally reinforced, etc., composites containing different (large and small) amounts of fillers and plasticizers.     

Analyzing of the effect of the dusty fluid,anisotropy and layered of the wall on the wave propagation

This study considers the propagation of time harmonic waves in, prestressed, anisotropic elastic tubes filled with viscous fluid containing dusty particles. The fluid is assumed to be incompressible and Newtonian. The tube material is considered to be incompressible, anisotropic, and elastic. The tube is subjected to a static inner pressure P_i and an axial stretch λ. Utilizing the theory of “Superposing small deformations on large initial static deformations”, differential equations governing wave propagation inside the tube are obtained in terms of cylindrical coordinates. Analytical solutions for the equations of motion for the dust and the fluid are obtained, and expressed numerically. The dispersion relation is obtained as a function of the stretch, the thickness ratio and the parameters for dusty particles.     

3E系统协调发展的PLSPM-GIA测度模型及其应用研究

在总结协调发展水平测算时应该注意的一些问题的基础上,提出建立PLSPM-GIA模型对3E系统协调发展水平进行估算。首先设计3E系统综合发展指标体系,运用PLS路径模型对中国1996-2014年的各系统的综合发展水平进行了计算,然后利用灰色关联模型度量了系统协调度,最后综合考虑系统的发展水平和协调水平,得出系统的协调发展水平。研究表明:①PLSPM-GIA模型能够弥补现有方法的缺陷,而且适用于系统协调发展水平的测度;②从发展水平来看,由高到低依次为经济子系统、能源子系统和环境子系统;③从系统的协调度来看,2005年之前能源经济子系统和能源环境子系统的协调度先上升后缓慢下降,由于受到经济环境子系统协调度缓慢下降的影响,3E系统协调度呈缓慢下降态势,2005年以后3E系统及其子系统的协调度下降趋势明显;④从协调发展水平来看,目前除经济环境子系统外,3E系统和其他子系统均处于轻度失调状态。