差分方程

The control principle of electrostatic shaping was introduced according to the balance between electrostatic force and resulting force formed by membrane deformation and the complex shaping process of SMEC. Then, taking the trisection circularity electrode for an example, the distribution characteristic of electric potential in the electrostatic field was analyzed, namely, the expression of potential function in the electrostatic field was deduced by Laplacian equation. And then, by combining the difference equation with electric potential expression, the numerical solutions of electrostatic force in single electrode mode and trisection circularity electrode mode were disposed. Finally, the calculated figure was compared with the ideal paraboloid and comparison shows that more accuracy would be achieved by multi-electrode control.

根据静电力与薄膜变形载荷作用力之间的平衡关系和静电拉伸薄膜反射镜成形的复杂过程，介绍了薄膜反射镜静电成形的控制原理；以三等分环状电极为例，分析了静电场中空间电势分布特性，即从拉普拉斯方程推导出静态场势函数的表达式；然后，利用差分与电势方程结合的方法，对单电极电场力和三等分环状电极电场力进行了数值求解；最后，将计算面形与理想抛物面进行了比较，结果显示，单电极情况下得到的薄膜反射镜面形不是理想抛物面，若采用多电极控制可获得更高的控制精度。

The control principle of electrostatic shaping was introduced according to the balance between electrostatic force and resulting force formed by membrane deformation and the complex shaping process of SMEC. Then, taking the trisection circularity electrode for an example, the distribution characteristic of electric potential in the electrostatic field was analyzed, namely, the expression of potential function in the electrostatic field was deduced by Laplacian equation. And then, by combining the difference equation with electric potential expression, the numerical solutions of electrostatic force in single electrode mode and trisection circularity electrode mode were disposed.

根据静电力与薄膜变形载荷作用力之间的平衡关系和静电拉伸薄膜反射镜成形的复杂过程，介绍了薄膜反射镜静电成形的控制原理；以三等分环状电极为例，分析了静电场中空间电势分布特性，即从拉普拉斯方程推导出静态场势函数的表达式；然后，利用差分与电势方程结合的方法，对单电极电场力和三等分环状电极电场力进行了数值求解；最后，将计算面形与理想抛物面进行了比较，结果显示，单电极情况下得到的薄膜反射镜面形不是理想抛物面，若采用多电极控制可获得更高的控制精度。

In this paper, we constructed the new conservative schemes of regularized long wave equation according to the physical laws of this equation.

在本文中，我们基于RLW方程本身具有的物理律，构造RLW方程的新的守恒型差分格式。

Dinger equation becomes a standard Mathieu equation, and then the basal quantum character such as energy spectrum and wave functions of the system are obtained.

该系统的差分薛定谔方程在电流的表象中可以变为马丢方程的形式，从而系统最基本的量子性质——能级、本征态和电流的量子涨落可以在WKBJ近似下完成。

This thesis is divided into six parts. The first chapter is preface, the current status of research in the inverse problems for parabolic partial differential equations is reported; the second chapter is "regularization methods for numerical differentiation and their applications ", in this chapter we investigate many regularization methods from a viewpoint of regularization theory and algorithm, some applications in the inverse problems for parabolic partial differential equations are given; the third chapter is "spectral regularization methods". Based on Fourier analysis, within the framework of regularization theory, we apply the spectral methods to some ill-posed problems. Many numerical experiments are done in order to show the validity of the methods; the fourth chapter is devoted to wavelet dual least squares method and a revised wavelet method; in the fifth chapter,we combine finite difference method with method of lines and apply it to the backward heat conduction problem in time; in the sixth chapter "identification problems for unknown source ", the essence and the degree of two problems related to source identification are pointed out, at the same time, some numerical methods are reported.

本文分为六个部分，第一章前言简要分析了国内外抛物型偏微分方程反问题的研究现状；第二章数值微分的正则化及其应用从正则化理论和算法的角度出发，考察了许多正则化方法，还给出了数值微分在抛物型偏微分方程反问题的一些应用；第三章谱正则化方法是在Fourier分析的基础上，在一般正则化理论的框架下，给出了这种方法在各种不适定问题中的应用，数值实验表明谱方法是有效的；第四章研究了小波对偶最小二乘方法和改进的小波方法；第五章主要研究了有限差分方法结合线方法在时间反向热传导问题中的应用；第六章是未知源识别问题，主要指出了两类未知源问题的不适定程度和不适定本质，同时报告了一些数值方法。

The patial differential equations with singular coefficients are very important ones in the field of numerical mathematics.

具有奇异系数的椭圆及抛物偏微分方程是一类很重要的方程，最近十几年，计算数学工作者利用有限差分、有限元等方法对此类问题进行了深入的研究并得到了很好的结果。

Based on this, a new WENO difference scheme which based on Dispersion-Relation-Preserving relation is developed, and representative test cases with this scheme for computational aeroacoustics problems has been implemented and compared in order to test capability of wave capturing; In addition, WENO schemes generally do not converge at high order in the presence of contact discontinuity of Euler equations, so a conservative front tracking technique coupling WENO schemes and Level Set method to simulate the translating density profile is presented here, and numerical simulation with this technique for representative test case has been implemented and results show the desired accuracy.

本文研究了高阶精度加权基本无振荡格式及其在双曲守恒律方程中的应用，在此基础上作了两个方面的工作：一是针对高频声波问题构造出一种基于保色散关系的WENO有限差分格式，并对计算气动声学问题的代表性算例进行了大量数值实验，比较了该格式捕捉波数的能力；另外，针对高阶WENO格式在处理Euler方程的接触间断时精度有所降低的问题，研究了利用界面追踪技术Level Set方法和高阶激波捕捉WENO格式相结合的一种守恒追踪方法，并且给出有代表性的密度滑移面问题的算例，得到一致高阶精度的数值模拟结果。

First, the finite difference was used to solve numerically the coupled Navier-Stokes equation and the convective-diffusion equation for surfactant in a orthogonal body-fitted coordinate system. The effect of surfactant on the flow field inside and around the deformed drop was successfully simulated.

本文在正交贴体坐标系下采用有限差分方法，对不可压缩流体的Navier-Stokes方程和表面活性剂的对流—扩散方程耦合求解，从而首先模拟了表面活性剂对单液滴运动的影响。

The FFT of Helmholtz equation on the regular domain is studied. And the multifrontal algorithm is used to solve the matrix equations in computational electromagnetics.Finally, the finite-difference approximate forms of Maxwell equations and the Despres transmission condition are discussed.

为了充分提高算法的计算效率，研究了规则区域上Helmholtz方程的FFT快速算法，以及有效地将多波前算法引入计算电磁学领域用于求解差分稀疏矩阵方程。

If the step of the linear system question is not very high, direct solving method is better; if coefficient matrix of the linear system question is not regular and large-sparse matrix , direct solving method does not settle the storage question.

而在求解线性方程组的许多实际问题中，尤其在偏微分方程的差分方法与有限元方法求解问题之中，方程具有重要的特征，一是多为大型稀疏矩阵；二是满足一些条件如对称正定、对角占优等，这使迭代法得到广泛的应用。

It has been put forward that there exists single Ball point and double Ball points on the symmetrical connecting-rod curves of equilateral mechanisms.

从鲍尔点的形成原理出发，分析对称连杆曲线上鲍尔点的产生条件，提出等边机构的对称连杆曲线上有单鲍尔点和双鲍尔点。

The factory affiliated to the Group primarily manufactures multiple-purpose pincers, baking kits, knives, scissors, kitchenware, gardening tools and beauty care kits as well as other hardware tools, the annual production value of which reaches US$ 30 million dollars.

集团所属工厂主要生产多用钳、烤具、刀具、剪刀、厨具、花园工具、美容套等五金产品，年生产总值3000万美元，产品价廉物美、选料上乘、质量保证，深受国内外客户的青睐