Firstly, based on the method of pressure-velocity coupling introduced by SIMPLE algorithm and a revised method of solving the unsteady equation of stream function and vorticity on uniform grid, a special algorithm called primitive-variable-time-dependent approach is formed in the present study. In this algorithm the Navier-Stokes equation in primitive variable form is solved by using the time-dependent approach on non-uniform staggered grid system. The second-order accuracy finite difference is used for the first and second partial derivatives of all variables of the convective terms and diffusion terms.
首先，本文在SIMPLE算法系列处理速度与压力耦合问题的思路和均匀网格下改进的关于流函数涡量方程的数值求解方法的基础上，将非定常原始变量Navier-Stokes方程的求解推广至非等距交错网格剖分，形成了具有本文特色的原始变量时间推进方法，其中对流项和扩散项中各变量的各阶导数均采用二阶精度公式，包括温度在内的离散方程组采用ADI迭代方法求解。
Within this context, four specific areas are addressed:(1) By means of finite integration technique, a new kind of the first order partial difference equation is derived from the original disperse transmission line equation of the uniform waveguide's. As it is the kind of one dimension Dirichlet's boundary problem, it is convenient for us to solve this equation from the leapfrog scheme. Because computation is carried out in one dimension, both high calculation efficiency and precision have been obtained in this method. Meanwhile, this method provide us a different selection to simulate the transient response of waveguide with non-simplical, for examples cylinder and elliptic waveguide, and avoid solving the second order equation, or using finite difference time domain to simulate a three dimension problem, sometimes the latter precision is not satisfied with the need, or low efficiency.
在这一研究内容下，主要研究四个方面的问题：(1)在完成金属波导传输线方程时域形式的基础之上，应用有限积分技术，把波导特征模式的色散传输线方程，化简为一组新的一阶偏微分方程组，该边值问题属一维Dirichlet边值问题，从而便于用蛙跳格式求解，由于是在一维中计算，该方法具有很高计算效率和精度，从而避免了以往为得到金属波导中特征模的时域响应特性，须要求解二阶方程，或用时域有限差分方法求解三维问题的方法，对于后者来说，计算有时是不准确的，或是很耗时的例如计算诸如圆波导、椭圆波导等其它复杂形状的波导。
For more and more people buy houses and cars on mortgage loan in modern society, we build the mathematical models of average monthly installment and average degressive corpus installment by using dynamic system model and the first order linear difference equation, and with the real example to show its usage in our life, meanwhile it giving us the reasons of differences between this two arithmetic.
针对当前社会上流行的按揭贷款买房、购车等现象，利用动力系统模型与一阶线性差分方程，分别建立了月均等额还款与等额本金递减还款法的数学模型，并结合实例说明其在经济生活中的具体运用，同时还给出了两种算法所产生差额的原因所在。
When the Burgers equation has periodic boundary conditions, first of all, we construct a class of alternating group four-point difference method using these four high-order alternating segment explicit and implicit non-symmetric form, and we can get the corresponding matrix format.
当Burgers方程具有周期性边界条件时，首先利用这四种高阶非对称格式构造了一种交替分组四点格式，并得到相应的矩阵形式。
Replacing the central difference operator with finite first order difference operator (Crank-Nicolson), we can get 45 ° one-way wave equation in unconditionally stable difference scheme.
本文在该方程中进一步引入了Crank-Nicolson一阶有限差分算子来代替中心差分算子，从而得出了45°无条件稳定差分格式的单行波方程。
Semi-difference method for solving first boundary value problem of 2nd order quasi-linear degenerate parabolic equation
解二阶拟线性退缩抛物方程第一边值问题的半差分方法
The differential difference equation described the optical bistability with delayed feedback is reduced to a difference equation with delayed time T + τ in the first-order approximation, but to a multiple iterated equation in the higher-order approximation.
在一阶近似下，描述延时反馈光学双稳性的差分&微分方程被简化为一个延时为T+τ的等效延时差分方程，在进一步的近似下被简化为一个多重迭代方程。
In this paper, an implicit-explicit upwind difference scheme for the first order linear hyperbolic equation is proposed using explicit scheme and implicit scheme alternately.
本文采用显格式与隐格式交替使用的方法，针对一阶线性双曲方程提出了一种隐-显迎风差分格式。
This kind of cyclic tridiagonal Toeplitz systems occurs repeatedly in the solution of implicit finite difference equations derived from linear first order hyperbolic equations, i. e. the transport equation, under a variety of boundary conditions.
在求解由一阶线性双曲型方程（如迁移方程）在一定边界条件下导出的隐式差分方程组时，要重复地求解此类Toeplitz循环三对角方程组。
By using the reducing order method and the general solution of first order constant coefficient linear difference equation, a general solution form of second order constant coefficient linear difference equation is derived.
利用降阶法及一阶常系数线性差分方程的通解，推导出二阶常系数线性差分方程的通解形式。
The oscillation of first order neutral delay difference equations with constant coefficients is studied by using characteristic equation method and a oscillatory criteria for the equation is established and at the same time, we expands this conclusion to the cases with several delays.
先是用特征方程法研究了一阶常系数中立型时滞差分方程的振动性并建立了方程的一个振动准则，并把该结果推广到多时滞的情形。
First, using some transformations, the authors obtain a conservation form from the Eikonal equation, and then not only give two types of finite difference schemes which have first order and second order accuracy respectively, but also prove that they are conditionally convergent.
首先通过变换，将程函方程化成一守恒形式；然后再差分离散，得到一阶和二阶精度的差分格式，并证明了差分格式条件稳定。
In this paper, the first order generalized difference scheme for linear Sobolev equation in two-dimension is considered.
本文考虑了二维线性Sobolev方程的一阶广义差分法。
Making use of the fourth-order compact difference formula of the first derivative term, anew high accuracy difference scheme for the two-dimensional convection reaction equation is proposed.
利用一阶偏导数项的四阶紧致差分算子，直接推导出了数值求解二维对流反应方程的一种新的高精度紧致差分格式。
This kind of cyclic tridiagonal Toeplitz systems occurs repeatedly in the solution of implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the transport equation, under a variety of boundary conditions.
在求解由一阶线性双曲型方程（如迁移方程）在一定边界条件下导出的隐式差分方程组时，要重复地求解此类Toeplitz循环三对角方程组。