数值微分

Keywords: Delay differential equation;Numerical method;Numerical stability

中文关键词：延时微分方程；数值方法；数值稳定性

Based on initial condition, during certain time span, numerical analysis method of differential equation, algebra iteration and optimization method are used to solve response of generalized system and to analyze stability of generalized system.

从初始条件出发，在一定时间长度上，采用微分方程数值积分法、代数迭代和优化方法来求解系统响应，获得非线性广义系统稳定性的数值分析方法。

In this paper, we construct the differential equation to solve the unconstrained optimization problem. Such methods solving the ill-conditioned problems are very effective. It indicates that the asymptotically stable region of explicit the Euler method to solve the differential equation equals to the descent region of the gradient method to solve the original unconstrained optimization problem.

我们发现，在用Euler方法求解二次优化问题的等价动力系统的方程时，由方法的步长确定的稳定区域对应于这些方法所得到的迭代公式的步长满足单调下降算法的条件确定的单调下降区域，因此我们可以利用这个性质构造解无约束优化问题的数值方法而不采用标准的常微分方程的数值求解公式。

The GKS stability theorem for one-dimensional model of hyperbolic initial and boundary value problem, Trefethen's explanation and extension based on the concept of group velocity are generalized firstly. Another possible high-frequency instability mechanism in numerical realization of MTF is pointed out based on his explanation and extension, namely, coupling effect of outgoing harmonic wave along one direction and node motion in the other directions can reverse the energy propagation in multi-dimensional discrete grids.

本文概述了—维双曲型偏微分方程组初边值问题数值稳定性的GKS定理及Trefethen基于群速度概念对这一定理的解释和推广；并应用Trefethen的解释和推广指出了MTF在数值实现中可能出现的另一种高频失稳机制，即在多维离散网格中，沿某一空间方向的外行简谐波，由于与其他空间方向节点运动的耦合效应可以使能量传播方向反向。

Moreover, some results about the stability of different numerical methods for the delay system are also itemized.

进一步，又对各种计算方法的数值稳定性进行了回顾，概述了各种常用的数值方法求解延迟微分方程所取得的理论结果。

However, the multi-time values before the initial time are required in solving the ordinary differential equation with linear multi-step method, the values at these time levels are still computed from the given initial value by one of above mentioned schemes.

在常微分方程的线性多步法中用到初始时次前若干时次的数值，但这些时次上的起动值仍然由初值运用方程计算得到的。这就是说，人们在预报问题中始终恪守只用一个初值的原则。在天气预报中，数值积分只用到一个起报场，即最后一个时次的观测场。

This paper considers the pricing model of options under illiquidity. A new numerical procedure for solving the nonlinear parabolic partial differential equation is explored and the Thomas algorithm is used to improving the efficiency of the numerical scheme.

本篇论文提出一个在市场流动性不足情况发生时的选择权订价模型，并且发展一个新的数值方法来求解一个非线性抛物线型态的偏微分方程式，同时利用汤马斯演算法来提升数值运算的效率。

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。

In the second chapter, generalizing the contractilities and asymptotical stabilities for multi-delay integro-differential equations. Under proper stepsize, we obtain the discretization schemes of Runge-Kutta methods with the compound quadrature formula and the Pouzet quadrature formula, and besides, derive the global and asymptotical stabilities. Moreover, the numerical experiments show that the presented methods are highly effective.

第二章，考虑了多延迟情形下的积分微分方程，推广理论解的收缩性与渐近稳定性结果，在适当的步长下，利用复合求积公式与Pouzet求积公式扩展Runge-Kutta方法，获得离散的计算格式，并且证明了方法是数值稳定的，此外，数值试验表明此计算方法在实际应用中是非常有效的。

The value solution adopts "the step-type" format to solve separate lights for the approximate value. The Euler method and the Runge—Kutta method are the most typical commonly value method to solve the differential equation.

数值解法采取"步进式"的方式求解离散点上的近似值，欧拉方法和龙格——库塔方法是求解微分方程最典型常用的数值方法。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新