边界条件

We deal with the biharmonic equation and proved that under our assumptions, the week solution of the natural boundary value problem strongly converges (in H~2) to that of the first boundary value problem.

为克服这一困难，我们用具自然边界条件的边值问题(在二阶椭圆型方程情形为第三边值问题)去逼近相应的第一边值问题，并以重调和方程为例，证明了在一定条件下，其具自然边界条件的边值问题的弱解在H~2的意义下强收敛到相应的第一边值问题的弱解。

To research the propagation characteristics of travelling wave along transmission line in depth, the authors establish a mathematical model of travelling wave along lossless transmission line with the format of discrete time and space mesh lines, derive the stable condition and boundary condition of the wave equations using numerical solution of partial differential equation, and attempt the simplification of the boundary condition to realize full decoupling of voltage and current.

为深入研究行波在配电线路上的传播特性，建立了时间-空间网格线离散格式的无损传输线行波数学模型，采用偏微分方程的数值解法推导了波动方程的稳定条件和边界条件，并尝试对边界条件进行简化，以实现电压和电流的完全解耦。

What differ from that of the researchers in the past is the consideration of the direction of Hopf bifurcation with Dirichlet boundary condition , that is, the conditions when the Hopf bifurcation periodic solutions are orbitally asymptotically stable with asymptotic phase.Our basic idea come from the results given by Tang[30]. We adopt the method used by Hassard in the Hopf bifurcation theorem in [24], which is to judge the existence and the direction of Hopf bifurcation, and the stability of the Hopf bifurcation periodic solutions.

与以前的研究者不同的是，本文不仅讨论了在Dirichlet边界条件下系统的Hopf分支在以扩散系数为分支参数的情况下的存在性和分支周期解的空间非齐次性，而且更多的关注了系统在Dirichlet边界条件下Hopf分支解的稳定性及Hopf分支的方向，也即系统定态解在何种条件下，当分支参数如何变化时产生Hopf分支且分支周期解是带渐近位相轨道渐近稳定的。

Chapter 2 and Chapter 3 of this dissertation,we discussoptimal boundary control problems for a semilinear elliptic type equation with linearboundary condition and nonlinear boundary condition,respectively.

全文共分为三部分内容：在第一部分即本文的第二章和第三章中，我们分别讨论了带有线性边界条件以及非线性边界条件的半线性椭圆型方程的最优边界控制问题。

LBIE, based on the local boundary equation, adopts the traditional moving least squares approximation which depends on only the values of the nodes in the domain of the problem or along its boundary. The whole process of integration is carried on over a local domain or its local boundary centered at the node in question. The local boundary equation can be rewritten to represent the values of the unknown function at the point of interest, and the essential boundary conditions can be directly and easily enforced by using the Green formula and the characters of the Dirac function.

它以局部边界积分方程为基础，采用移动最小二乘近似函数，从而只需要分布在问题域内及其边界上的节点的信息值，无需划分单元；整个积分是在以节点为中心的局部域及其边界上实现，所以不需要背景积分网格；借助于格林公式及Dirac函数的性质，将局部边界积分方程转化为所考虑点的未知函数的边界积分表达式，便于直接施加本质边界条件。

Corresponding mathematics model was developed, hole-edge stress analysis on composite material plate with multiform holes was carried out, accurate boundary conditions was founded by conformal mapping method, boundary problems of the two stress functions could be treated by affine transformation in the same way synchronously.

建立了相应的数学模型，对含不同孔型复合材料板进行了孔边应力分析，通过保角映射方法建立精确的边界条件，解决了复杂孔型的边界条件问题，借助仿射变换能同时并且同方法的处理这两个应力函数在边界上的问题。

The free-surface boundary conditions are satisfied on the free-moving surface exactly without any approximation or linearization. The governing equations and the boundary conditions on the time-dependent, physical domain are transformed onto a regular domain for numerical computations.

为於未知的自由液面边界上满足非线性边界条件，本模式利用座标转换，将非稳态且不规则物理区间的流场控制方程式与边界条件，映射至规则的计算区间，以进行数值解析。

The governing equation is the mild-slope equation. Finite element method has advantage to treat complicated boundary. The second order radiation boundary condition is expanded to improve the wave model for large incident wave angles. The validity of the second order radiation condition is examined through varying arbitrary wave incident angles.

模式将以包含波浪绕射及反射之缓坡方程式建立数值模式，利用有限元素法易於处理不规则边界的优点，引用二阶近似辐射边界条件，扩展波场模式处理大角度波浪入射的能力，探讨在未碎波的情形下，二阶辐射边界条件对於任意角度波浪入射之适用情形。

As same as the above-mentioned reason, because theconditions at the inside straight boundary of the binarycomposite material can not be easily satisfied, here, anelastic circular inclusion with large radius is used to replacethe upper half space of the binary composite material, so the〓mary binary composite mater ial problem can be changed〓the scattering problem of a circular elastic inclusion andto the scattering problem of a circular eastic inclusion and〓circular cavity in an infinite space to the steady incident〓wave.

与前面两个问题的理由一样，同样，由于问题中双质复合材料的内部直边界上的边界条件不易解析满足，为此，本文采用了一个半径很大的弹性实夹杂来代替问题中双质复合材料的上半空间部分，而将原来的双质复合材料问题转化成了无限大空间中一圆形弹性夹杂和一圆孔对稳态P波的共同散射问题，根据该问题的特点写出了弹性夹杂内外波场中波函数的一般形式解，利用该问题中夹杂和圆孔处的位移和应力的连续条件，并采用复数傅立叶级数展开技术将其化为一个仅包含问题波函数中未知系数的无穷线性代数方程组，在满足一定计算精度的前提下，通过有限项截断进行近似求解，进而讨论了双质复合材料中圆孔边界处的动应力集中系数随不同入射角、圆孔的不同位置、以及入射波的不同无量纲波数的变化和分布情况。

An ABCs based on scalar magnetic potential is proposed and the correspondent mathematical models in spherical boundaries are established.

首先提出了基于标量磁位的渐近边界条件，建立了球形边界的标量磁位渐近边界条件数学模型。

He is the most remarkable man that I have ever known.

他是我知道的人中最值得注意的人。

Increased appearance of lymphocele was noticed in patients with diabetic nephropathy, congenital malformations of the urinary tract, and inflammatory diseases, including glomerulopathy and extraglomerular ones, after high-voltage radiotherapy and after removal of the renal graft.

经过高压放射疗法和切除移植肾后，在糖尿病肾病患者、泌尿系统先天性畸形患者和炎症患者身上发现淋巴囊肿有所增加，包括肾小球病和球外疾病等。