方程的解析与欣赏

一、欣赏方程中的文化 含有未知数的等式叫做方程．研究方程必须研究等式,关注方程与等式的区别与联系．香港数学教育专家提出一个以下与正常思维相悖的问题,会使学生感到震惊．     

十一、参数方程与极坐标

知识要点］本章的主要考试内容：１．曲线的参数方程,参数方程与普通方程的互化．２．极坐标系,曲线的极坐标方程,圆锥曲线的极坐标方程,极坐标与直角坐标的互化．［地位和作用］参数方程是曲线方程的一种表现形式,它借助于函数、方程、不等式、三角等研究方法和研究...     

关于发展方程的守恒率的一个标记

本文使用微分代数的技巧，研究了发展方程的守恒率与对应的行波所满足方程的首次积分之间的关系．通过文本给出的结果，我们研究了Burgers方程和Burgers—KdV方程的可积性，证明了这两类方程都只有一个守恒率．利用本文给出的方法，可以通过常微分方程的研究方法来研究某些非线性发展方程．     

Kac方程的解及其性质

研究Kac方程的初值问题．证明了该类方程存在唯一的全局分布解．并且使用一种新的线性化方法证明了该类方程的解具有相应的多项式衰减性．     

“直线方程的一般式”教学设计

一、教材分析 本课所用的教材是苏教版必修2的第二章2．1．2直线方程的一般式． 1．教材的地位 直线方程的一般式是继“直线方程的四种特殊形式”之后,对直线方程作进一步研究．从教材整体看：直线的方程既是初中二元一次方程知识的延续（数形结合）,     

十、圆锥曲线

知识要点］曲线与方程的概念．能利用给定条件选择适当坐标系求出曲线的方程，能通过曲线的方程来研究曲线的性质．对于一些常见曲线的方程应能根据其方程画出该方程所表示的曲线．理解充要条件的意义及在数学变形过程中的等价性问题．圆锥曲线的定义、它们的标准方程及有...     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

强阻尼非线性波方程的近似惯性流形

本文研究一类强阻尼非线性波方程．通过交换的方法，我们构造了方程的近似惯性流形序列，并得到了对于方程的整体吸引子的逼近估计．     

Burgers方程的混合元分析及其数值模拟

1．引言混合有限元法在高阶偏微分方程和含有两个战者两个以上）的未知国数的偏微分方程的数值解的研究中起着重要的作用．但是,到目前为止,混合有限元法主要是用于2n阶或一阶偏微分方程（组）,如二阶椭圆型方程、平面弹性力学方程、双调和方程、Stokes和Navier－stokes方程、抛物型方程以及电磁场方程修见＞到以及当中的参考文献）．然而,R前混合有限元法还没有被用于对非线性的Burgers方程作数值研究．而过去对Burgers方程的数值研究主要采用标准有限元法、差分方法和谱方法修见【IO－12］以及当中的参考文献）．本文的目的是用混…     

关于Dufing方程周期解的存在性与唯一性

采用Lyapunov-Schmidt约化结合Sturm特征值比较方法,将二阶Duffing方程转化为单个函数方程．通过研究函数的单调性,给出Duffing方程的周期解的存在性与唯一性．     

An achievement rate approach to linear programming problems with an interval objective function

In this paper, we focus on a treatment of a linear programming problem with an interval objective function. From the viewpoint of the achievement rate, a new solution concept, the maximin achievement rate solution, is proposed. Nice properties of this solution are shown: a maximin achievement rate solution is necessarily optimal when a necessarily optimal solution exists, and if not, then it is still a possibly optimal solution. An algorithm for a maximin achievement rate solution is proposed based on a relaxation procedure together with a simplex method. A numerical example is given to demonstrate the proposed solution algorithm.     

Global efficiency in multiobjective programming *

In this paper, we are concerned with global efficiency in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove that a local weakly efficient solution, a local efficient solution and a local properly efficient solution are respectively a global weakly efficient solution, a global efficient solution and a global properly efficient solution of a multiobjective programming problem if cone- subconvexlikeness or cone-pre-invexity is assumed     

On fourth-order elliptic boundary value problems with nonmonotone nonlinear function

This paper is concerned with the fourth-order elliptic boundary value problems with nonmonotone nonlinear function. The existence and uniqueness of a solution is proven by the method of upper and lower solutions. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution.     

Periodic solutions in n dimensions and Volterra equations

If a nonlinear autonomous n-dimensional system of ordinary differential equations has a bounded solution with a certain uniform stability property, this solution approaches an almost periodic solution with the same stability property. (More precisely, the almost periodic solution is in the set of ω-limit points of the given solution.) If the bounded solution has, in addition to the uniform stability property, an asymptotic stability property, then the solution approaches a periodic solution with the same stability properties. Practical (i.e., computable) sufficient conditions for boundedness of solutions are obtained. The results are applied to generalized Volterra equations.     

向量均衡问题的超有效性的对偶形式

在局部凸空间中引进了向量均衡问题的强超有效解、C-强超有效解、弱超有效解, C-弱超有效解、齐次超有效解、 C-齐次超有效解的概念,并在局部凸空间中用极理论为工具讨论了向量均衡问题的 C-弱超有效解, C-超有效解, C-齐次超有效解,以及C-强超有效解的对偶形式. 又在赋范线性空间中讨论了向量均衡问题的以上各种超有效解之间的等价性,并且在赋范线性空间具正规锥的条件下讨论了向量均衡问题的以上各种超有效解的对偶形式. 作为它的应用,给出了向量优化问题各种超有效解的对偶形式.     

Convolution solutions of an abstract stochastic Cauchy problem

We study convolution solutions of an abstract stochastic Cauchy problem with the generator of a convolution operator semigroup. In the case of additive noise, we prove the existence and uniqueness of a weak convolution solution; this solution is described by a formula generalizing the classical Cauchy formula in which the solution operators of the homogeneous problem are replaced by the convolution solution operators of the homogeneous problem. For the problem with multiplicative noise, we find a condition under which the weak convolution solution coincides with the soft solution and indicate a sufficient condition for the existence and uniqueness of a weak convolution solution; the latter can be obtained by the successive approximation method.     

Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is polynomial, then the exact solution can be obtained. If the exact solution is not a polynomial, then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver. In addition, the method is presented with error and stability analysis.     

Asymptotic Behavior of the Solution to a 3-D Simplified Energy-Transport Model for Semiconductors

The well-posedness of smooth solution to a 3-Dsimplified Energy-Transport model is discussed in this paper. We prove the local existence, uniqueness, and asymptotic behavior of solution to the equations with hybrid cross-diffusion. The smooth solution convergences to a stationary solution with an exponential rate as time tends to infinity when the initial date is a small perturbation of the stationary solution.     

A new analytical solution branch for the Blasius equation with a shrinking sheet

A similarity equation of the momentum boundary layer is analytically studied for a moving flat plate with mass transfer in a stationary fluid by a newly developed technique namely homotopy analysis method (HAM). The equation shows its significance for the practical problem of a shrinking sheet with a constant velocity, and only admits the existence of the solution with mass suction at the wall surface. The present work provides analytically new solution branch of the Blasius equation with a shrinking sheet in different solution areas, including both multiple solutions and unique solution with the aid of an introduced auxiliary function. The analytical results show that quite complicated behavior with three different solution areas controlled by two critical mass transfer parameters exists, which agrees well with the numerical techniques and greatly differs from the continuously stretching surface problem and the Blasius problem with a free stream. The new analytical solution branch of the Blasius equation with a shrinking sheet enriches the solution family of the Blasius equation, and helps to deeply understand the Blasius equation.     

Global solution to the incompressible flow of liquid crystals

The initial–boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution must be equal to the unique strong solution with the same data.