变分方程

The control volume integration was applied to deduce the discrete expressions of the convection-diffusion equations. The staggered grid and SIMPLE algorithm were introduced to deal with coupling between pressure and velocity, and then the numerical computation expressions of such variables as fluid flow velocity and pressure were deduced. Using the backward difference method and incremental theory to discretize the governing equations for fields of chemical reaction, material structure and chemorheology, the numerical computation expressions of variables such as the monomer conversion, average molecular weight and fluid viscosity were constructed.

应用控制容积积分法导出了对流—扩散方程的离散表达式，引入交错网格技术与SIMPLE算法，实现了耦合的压力场与速度场的分离式求解，推导得到了流体的流动速度、压力等物理量的数值计算式；采用向后差分方法和增量方法，实现了化学反应场、材料结构场、化学流变场控制方程的离散，获得了反应转化率、聚合物平均相对分子质量、流体黏度等物理量的数值计算式。

Continuity equation and motion equation which can be applied to cone-shaped pipes are obtained and finite difference method for them is brought forward.

根据非稳定流的一维流动理论，对圆锥形管道中的瞬变流动进行了分析，推导出了适用于圆锥形管道的连续性微分方程和运动微分方程，给出了这两个方程的有限差分解法，同时对计算结果进行了相对误差的对比分析。

Because the equation is a various coefficients second-order parabolical partial differential equation,explicit difference formats usually have high condition for stability. Implicit difference formats have good stability,but they must calculate a set of algebraic equations,which are too complex.

由于泥沙扩散方程是一个二维变系数的抛物型方程，对于多维的方程，显格式往往稳定性要求很高，而隐格式虽然稳定性较好，但它要解大型的代数方程组，工作量也非常大，为了克服这两种格式的不足，常用ADI法求解，但仍存在一些计算量偏大等不足之处，因此构造出更好的差分格式是有必要的。

Use of the finite difference method for Volterra equation here as a variable element of the equation, As the same...

用差分方法求解沃尔泰拉方程，此处为一个变元的方程，由于该方程同时含有微分和积分，一般求解有一定的困难。

Therefore,it is necessary to research diffusion equation for suspended sediment because it describes the sediment move process in the water body.The equation is a various coefficients second-order linear partial differential equa-tion,such equation under complex boundary condition is very difficult to get its analytical solution,while its numerical solution relative analytical solution is more easier and has the obvious superiority:simple,the computation convenience.but to get a kind of difference format which is good accuracy and stability is not easy.

泥沙扩散方程实际上是一个变系数的二阶线性偏微分方程，这样的方程在各种复杂边界条件下求解是十分困难的，求它的解析解在数学上存在着难以克服的障碍，无法求出其精确解，因此常用数值方法求它的近似解，相比较而言，数值方法有着明显的优势：即简单灵活、计算方便快捷，但要寻找一种精度高、稳定性好、计算方便的差分格式也并非易事。

The nonlinear hyperbolic conservation laws of the transient flow of the media in gas pipeline are hard to he solved, so, the control conservation laws are constituted of mass, momentum arid energy conservation laws, TVD/Godunov hydrid algorithm is used to simulate the transient flow in gas pipeline, and the traditional splitting approach processing nonlinear convection in momentum equation is dismissed ,the convection tern; is directly processed, in the way, the stablization of the algorithm and emulating accuracy of the transient flow are greatly improved.

针对管道中介质瞬变流动的非线性偏微分方程组难以解析求解的问题，由质量、动量、能量三个偏微分方程构成的控制方程组，应用TVD/Godunov混合算法数值模拟天然气管道中气体的瞬变流动，摒弃了传统的差分方法时动量方程非线性时流项的线性化处理，而直接处理非线性对流项，从而大大提高了算法的稳定性和管道瞬变流动的仿真精度。

Considering of helical gear separate mesh and tooth-back shock, the non-linear vibration equitions of the system are also established, and the non-linear stable cycle solutions of the equitions are evaluated through dividing the timechanging stiffness matrics and dimension the equitions with selfadapt vary-step auto-RK4 numerical method.

考虑了斜齿轮传动的轮齿脱啮和齿后侧冲击，建立了系统的非线性振动方程，并通过对方程时变刚度阵的分块处理及无量刚化，用自适应变步长auto-RK4数值方法求出了系统非线性振动方程的稳态周期解。

The variational form of the governing equations is given on the basis of the extended Hu-Washizu three-field variational principle.

基于胡海昌-Washizu三变量广义变分原理结合特征线法给出了控制方程的单元弱形式。

The weak form of the governing equations of coupled thermo-hydro-mechanical problems in porous media within the element is established on the basis of the Hu-Washizu three-field variational principle.

基于胡海昌-Washizu 三变量广义变分原理给出的多孔介质中热-渗流-力学耦合问题控制方程的单元弱形式，导出了单元公式。

In Chapter 2, an existence theorem is obtained for periodic solutions of non-autonomous second order systems with even-typed potentials. We also investigate solutions of linear bounded positive self-conjugation operators equations in Hilbert Space by variation methods.

第二章主要利用临界点理论中的归药方法、极小作用原理研究具偶型位势的二阶非自治系统周期边值问题，得到了周期解的存在；利用变分方法证明了Hilbert空间中有界线性正自共轭算子方程解的存在性。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。