条件方程

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。

In Section 1, without assuming operators to be continuous and cones to be normal, we use the Monchs compactness condition to study the existence and iterative approximation theorem of solutions and coupled minimal and maximal quasi-solutions for a class of nonlinear operator equations x = Ax, x under more extensive and weaker conditions than those used in [18] in ordered Banach spaces.

践一节中，利用M加ch的紧性条件，在比文献［18］更广泛的较弱的条件下，在不假潞子连续和拄酬的前提下，研究了Ball''crl空间中一描F线腑子方程X—川巳F）的解和最小最大耦合解的存在与迭代跳定理，并应用到Ban呐空间中非线性VOlterra型积分方程和常徽分方程的初值问题。

For the Riemann boundary value problems for the first order elliptic systems , we translates them to equivalent singular integral equations and proves the existence of the solution to the discussed problems under some assumptions by means of generalized analytic function theory , singular integral equation theory , contract principle or generaliezed contract principle ; For the Riemann-Hilbert boundary value problems for the first order elliptic systems , we proves the problems solvable under some assumptions by means of generalized analytic function theory , Cauchy integral formula , function theoretic approaches and fixed point theorem ; the boundary element method for the Riemann-Hilbert boundary value problems for the generalized analytic function , we obtains the boundary integral equations by means of the generalized Cauchy integral formula of the generalized analytic function , introducing Cauchy principal value integration , dispersing the boundary of the area , and we obtains the solution to the problems using the boundary conditions .

对于一阶椭圆型方程组的Riemann边值问题，是通过把它们转化为与原问题等价的奇异积分方程，利用广义解析函数理论、奇异积分方程理论、压缩原理或广义压缩原理，证明在某些假设条件下所讨论问题的解的存在性；对于一阶椭圆型方程组的Riemann-Hilbert边值问题，利用广义解析函数理论、Cauchy积分公式、函数论方法和不动点原理，证明在某些假设条件下所讨论问题的可解性；广义解析函数的Riemann-Hilbert边值问题的边界元方法是利用广义解析函数的广义Cauchy积分公式，引入Cauchy主值积分，通过对区域边界的离散化，得到边界积分方程，再利用边界条件得到问题的解。

The phase of optical wave can be recovered by the intensity transport equation under the paraxial approximation For the optical system with diffraction limit, it is difficult to get the boundary radical slope of phase and sample in the circular domain boundary when using the intensity transport equation to recover the phase, consequently, we reviewed an algorithm mentioned in the reference ［4］, namely, the equation, computation domain and boundary condition are changed, then the phase is reconstructed by the multigrid method and corrected at last Moreover, the experiment system is set to testify the algorithm, and the recovered phase from the intensity data detected by CCD is compared with the result from phase retrieval algorithm, it is found that the phase reconstruction error can reach 017 λ in the case of non-uniform intensity distribution, as a result, the method is suitable for the phase recovery with low wavefront sensing accuracy.

在傍轴近似条件下，可以利用光强传播方程对畸变波前进行相位恢复。对于衍射受限的光学系统，很难获得相位的边界径向斜率值作为边界条件，此外，要获得精确的圆域边界采样值也并非易事。为了克服上述困难，进一步研究了相位恢复改进方法，即改变了方程的表示形式、计算区域和边界条件，并用多重网格法进行求解获得重构相位，最后再将其修正。为了验证算法的精确性，搭建了实验系统对算法进行实验验证，将由CCD探测的光强分布进行计算得到的相位分布与相位恢复算法的结果进行比较，证明在光强分布非均匀的情况下，该方法恢复的相位均方根误差能够达到017 λ，可以适用于波前传感精度要求不是很高的相位恢复。标签自适应光学相位恢复相位修正光强传播方程 adaptive optics phase retrieval phase correction intensity transport equation

At first,the cobweb model in a differential equation's form is established under the condition of continuous time.

首先，在连续时间的条件下，建立了蛛网模型的微分方程形式；其次，通过分析微分方程模型平衡点的稳定性，得到了蛛网模型稳定的条件；最后给出与微分方程模型相应的经济解释和具体应用。

Then the FPK equation governing the joint probability density of amplitude and phase, the backward Kolmogorov equation governing the conditional reliability function, and the Pontryagin equation governing the mean first passage time are established, respectively.

随机微分方程。在此基础上，建立联合概率密度满足的Fokker-Planck-Kolmogorov方程、条件可靠性函数满足的后向Kolmogorov方程以及平均首次穿越时间满足的Pontryagin方程，分别求解这些方程得到联合概率密度、条件可靠性函数以及平均首次穿越时间。

We can not apply Galerkin methods with radial basis functions to the first boundary value problem (the Dirichlet boundary value problem) of the second-order elliptic equation, because it requires that the approximate functions should fit the essential boundary condition, while the spaces spanned by the radial basis functions fails to satisfy this kind of boundary condition, and it is the same for the higher-order elliptic equation.

我们不能直接应用这样的无网格算法求解二阶椭圆型方程的第一类边值问题(Dirichlet边值问题)，因为第一类边界条件是本质边界条件(对二阶椭圆型方程为Dirichlet边界条件)，而径向基函数的支集通常是全空间，由它们所生成的近似函数空间一般不满足这样的条件。

The existence of global solutions of equations is the precondition to consider the asymptotic properties.

方程的整体解的存在性是考虑方程解的渐近性质的前提条件，线性增长条件是保证方程的整体解存在的最简单条件。

Relative to SDE, the study for the solution of BSDE under non-Lipschitz condition is absence, especially when the uniqueness of the solution can not be guaranteed, the existence of minimal and maximal solution of BSDE are not be studied.

相对于正向随机微分方程，非Lipschitz条件下倒向随机微分方程解的性质的研究尚不够丰富，特别是条件不能保证方程解唯一时，倒向随机微分方程最大最小解的存在性尚未见有成果。

Adding boundary conditions in whole equations in the wavelet Galerkin method frequently does with its of differential equations. The result can derive by solving transcendental equation, which make unknow quantity number and the number of system of equation become nonuniformity.

经常采用的小波Galerkin方法对微分方程边界条件的处理，均是将边界条件作为附加方程补充到整体方程中，从而求解超越方程组得到原方程的解，这导致了求解过程出现的方程组个数与未知量个数不一致。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。