定义方程

The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. The advantage is that the cost function variation is independent of the flow field variation, with the result that the gradient of cost function with respect to arbitrary number of design variables can be determined without the need for additional flow-field evaluations. So each design cycle requires the numerical solution of both the flow and the adjoint equations leading to a computational cost roughly equal to the cost of two flow solutions.

虽然他们的研究成果大多数都可以在公开性文献上看到，但是其中一些具体的细节问题的处理方法却没有透露，比如：伴随方程如何离散求解才能减小最终梯度计算中的误差，使得设计过程有效地进行；伴随方程与流动方程的边界条件是不同的，如何合理地处理伴随方程的边界条件，才能推动设计过程朝着设计目标而发展；在减阻问题中目标函数如何定义，求解梯度的具体数学公式如何推导，如何通过数值方法来实现它们等。

It uses a uses the MSFlexgrid to enter the augmented matrix but its limit is 253 because I use a Byte variable to define the equations number so if you want to use it for a bigger Linear System you will have to change some variable definitions

它使用一个MSFlexgrid 来输入参数矩阵，参数个数最多是253个，因为我使用一个Byte 变量来定义方程的未知数的个数。所以如果你想在更大的线性系统中使用它，你将必须修改变量的定义。

In the "Kinetics" hand-out see there the meanIn the "Kinetics" hand-out (see there the meanings of variables, the ings of variables, the Debye-Hückel Debye-Hückel equation is defined as:equation is defined as

在在""动力学动力学""讲义提纲中讲义提纲中(参考那里各变量的含义参考那里各变量的含义，， Debye-HückelDebye-Hückel方程被定义为方程被定义为

First of all,we have given some of the basic concepts of differential equations, described the constant coefficient linear ordinary differential equation solution, for a class of second-order variable coefficient linear ordinary differential equation initial value problem, an approximate solution, the method is first unknown function of a definition for N sub-interval, and then in between each district within a constant coefficient ordinary differential equations similar to the replacement, the solution has been the problem as similar to the original analytical solution, and then gives a detailed second-order change order coefficient of linear homogeneous ordinary differential equation solution examples, the examples of the approximate method proposed in this paper is valid.

首先给出了微分方程的一些基本概念，讲述了常系数线性常微分方程的解法，针对一类二阶变系数线性常微分方程初值问题，提出了一个近似解法，本方法是先对未知函数的一个定义区间作N等分，然后在每一个小区间内用一个常系数常微分方程近似替换，所得到的解作为原问题的近似解析解，随后详细给出了一个求二阶变系数齐次线性常微分方程的解的实例，该实例说明本文提出的近似方法是有效的。

On the whole, we do as follows: Firstly, we list some conceptions and lemmas for later use. Secondly, we define δ-fine partitions for infinite interval and integral of vector-valued functions on infinite interval, and discuss the properties of integral, and characterize its primitives...

主要包括以下五部分内容：在第一部分中，我们介绍了本文所用到的基本概念和引理；在第二部分中，通过定义无穷区间上δ-精细的分法，我们给出了无穷区间上向量值函数的积分的定义，并讨论其性质，还给出了原函数的刻划；在本文的第三部分中，我们着重讨论了无穷区间上向量值函数积分的收敛定理；在本文的在第四部分中，我们首先应用无穷区间上向量值函数积分的收敛定理给出了常微分方程整体广义解的存在性定理，其次应用强积分对Banach空间常微分方程广义解进行了讨论；最后，在第五部分中，我们将模糊积分推广到无穷区间上并给出了其数值计算方法。

The viscosity solutions for partial differential equations of parabolic type havethree different basic definitions.The three definitions are equivalent when the equa-tions are continuous.But for the discontinuous equations,the equivalence of themare still unknown.

因为抛物型方程粘性解有三种基本定义法，它们在方程为连续时相互等价，对于不连续情况，它们间关系尚不明确，因而在本章里，我们特别证明了其中二个定义：定义1，定义2之间的关系。

In the aspect of the study of the application of HHT method, This dissertation has done three part of jobs: first, the Elcentro strong ground motion which is in common use in earhtquake engineering is analyzed by HHT method, getting its amplitude-frequency-time distribution and the comparison between the HHT marginal spectrum and Fourier spectrum is also made; second, the typical nonlinear system defined by the Duffing equation is analyzed by HHT method, getting its numerical value's amplitude-frequency-time distribution and the comparison between the HHT marginal spectrum and Fourier spectrum is also made; third, define the stationary degree of non-stationary signals making use of 3-D time-frequency spectrum and marginal spectrum, the stationary degree curve of Elcentro seismic recording is protracted according to its defining formula, the stationary degree curve of Elcentro seismic recording which change with time is protracted according to similiar definition.

在HHT方法的应用研究方面，本文做了三部分的工作：首先，利用HHT方法分析了地震工程中常用的Elcentro非平稳强震记录，得到了该记录的振幅－频率－时间分布特性，并将得到的边际谱与傅立叶谱做了比较；其次，应用HHT方法分析了典型的非线性系统－Duffing方程，得到了该方程数值解的振幅－频率－时间分布特性，并将得到的边际谱与傅立叶谱做了比较；最后，利用三维时频谱与边际谱定义了非平稳信号的平稳度，并依照其定义式绘制了Elcentro地震波记录的平稳度曲线，依照类似与依赖与频率变化的平稳度定义，初步尝试编程绘制了Elcentro地震波随时间变化的平稳度曲线。

In this project, we study the theory of higher order differential equations in Banach spaces and related topics. We solve an open problem put forward by two American Mathematicians and two Italian Mathematicians concerning wave equations with generalized Weztzell boundary conditions, introduce an existence family of operators from a Banach space $Y$ to $X$ for the Cauchy problem for higher order differential equations in a Banach space $X$, establish a sufficient and necessary condition ensuring $ACP_n$ possesses an exponentially bounded existence family, as well as some basic results in a quite general setting about the existence and continuous dependence on initial data of the solutions of $ACP_n$ and $IACP_n$. We set up quite a few multiplicative and additive perturbation theorems for existence families governing a wide class of higher order differential equations, regularized cosine operator families, regularized semigroups, and solution operators of Volterra integral equations, obtain classical and strict solutions having optimal regularity for the inhomogeneous nonautonomous heat equations with generalized Wentzell boundary conditions, gain novel existence and uniqueness theorems,which extend essentially the existing results, for mild and classical solutions of nonlocal Cauchy problems for semilinear evolution equations, present a new theorem with regard to the boundary feedback stabilization of a hybrid system composed of a viscoelastic thin plate with one part of its edge clamped and the rest-free part attached to a visocelastic rigid body. Also we obtain many other research results.

在本研究中，我们对Banach空间中的高阶算子微分方程的理论以及相关理论进行了深入研究，解决了由美国和意大利的四位数学家联合提出的一个关于广义Wentzell边界条件下的波动方程适定性的公开问题，恰当地定义了Banach空间中的高阶算子微分方程Cauchy问题的算子存在族及唯一族，建立了齐次和非齐次高阶算子微分方程Cauchy问题适定性的判别定理，获得了关于高阶退化算子微分方程的算子存在族、正则余弦算子族、正则算子半群、Volterra积分方程解算子族的乘积扰动和混合扰动定理，得到了关于以依赖于时间的二阶微分算子为系数的一大类非自治热方程非齐次情形下的时变广义Wentzell动力边值问题的古典解、严格解的最大正则性结果，获得了半线性发展方程非局部Cauchy问题广义解和经典解存在唯一的判别条件，从实质上推广了现有的相关结果；得到了一部分边缘固定而另一部分附在一粘弹性刚体上的薄板构成的混合粘弹性系统的边界反馈稳定化的新稳定化定理，还建立了一系列其他研究结果。

To verify the accuracy of CFD simulations, experiments were carried out. The results of simulations with user defined function of the effects of impeller type and stirring rate on the velocity field were in good agreement with the output of particle image velocity. The oxygen mass transfer model can be used to predict the process of oxygen mass transfer in a vessel, and the logarithmical expression can successfully describe the relation between the oxygen concentration and the dissolution time.

结果表明，(1)采用Fluent软件并结合用户自定义方程(user defined function， UDF)能够很好地模拟出实际搅拌器内流场分布，模拟结果与采用粒子成像技术(particle image velocity， PIV)的实验测量结果相符；(2)采用氧气传质模型能预测氧气在搅拌器内的动态传质过程，同时氧气浓度与溶解时间的对数关系式能较好描述试验搅拌器内氧气动态传质过程；(3)在相同搅拌速度下，圆盘涡轮式搅拌器产生的湍流动能分布范围要大于桨式搅拌器产生的湍流动能，而且湍流动能分布更均匀，湍流强度更大。

In this paper,established the functions of irrigation application efficiency,irrigationdistribution efficiency and water store efficiency according to the parameters of water flowadvance and infiltration for furrow and border irrigation with tailor water condition.

本文提出根据膜孔沟灌的水量平衡原理，由水流行进方程和有关参数，确定入渗方程，并根据灌水均匀度、灌溉效率和储水效率的定义，推导出长沟畦灌溉有排水条件下由行进方程和入渗方程的参数确定的灌水均匀度方程，与沟畦长分段求解方法相比，其误差小于2%。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。