Numerical integration of systems of delay differential-algebraic equations

The numerical solution of the initial value problem for a system of delay differential-algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which ensures the best condition for the corresponding system of continuation equations. The best argument is the arc length along the integral curve of the problem. Algorithms and programs based on the continuous and discrete continuation methods are developed for the numerical integration of this problem. The efficiency of the suggested transformation is demonstrated using test examples.     

以函数为参量的间歇发酵非线性动力系统及其辨识

建立以连续分段线性函数为参量的间歇发酵非线性动力系统,证明该动力系统的主要性质及解的存在性.以实验数据拟合得到的光滑曲线为依据,提出了连续分段线性函数为优化变量的辨识模型,论述可辨识性.依状态变量与辨识函数的相关性,构造求解辨识模型的优化算法,并给出优化算法的收敛性分析及数值结果.     

一类带修理工休假可修系统解的存在唯一性及正则性分析

考虑一类修理工可多重延误休假的n部件串联可修复系统解的存在唯一性及正则性问题.通过将系统模型方程转化为一组算子积分方程,利用不动点理论讨论该系统局部解的存在唯一性问题,再由一致先验估计和连续延拓讨论系统整体解的存在唯一性问题,继而分析解的正则性问题.为解决复杂可修复系统解的存在唯一性及正则性提供了可行性方法,并且方法同样适用于排队论系统和其他类似系统.     

解对初值连续依赖定理的一个新的证明方法

《常微分方程》传统的教材均先证明Bellman引理,然后利用Bellman引理完成"解对初值连续依赖定理"的证明.本文用曲线平移的方法构造出比较函数,从而给出了解对初值连续依赖定理的一种新的且具有几何直观性的证明方法.     

On convergence of a discrete problem describing transport processes in the pressing section of a paper machine including dynamic capillary effects: One-dimensional case

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system of equations which describe the filtration process in the pressing section of a paper machine. Two flow regimes appear in the modeling of this problem. The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards’ approach together with a dynamic capillary pressure model. The finite volume method is used to approximate the system of PDEs. Then, the existence of a discrete solution to the proposed finite difference scheme is proven. Compactness of the set of all discrete solutions for different mesh sizes is proven. The main theorem shows that the discrete solution converges to the solution of the continuous problem. At the end we present numerical studies for the rate of convergence.     

导数振荡与应变能极小的平面分段三次参数Hermite插值

由于分段三次参数Hermite插值的切矢往往被作为变量,故可对其进行优化以使得构造的插值曲线满足特定的要求.为了构造兼具保形性与光顺性的平面分段三次参数Hermite插值曲线,给出了一种通过同时极小化导数振荡和应变能来确定切矢的方法.首先以导数振荡函数和应变能函数为双目标建立了切矢满足的方程系统;然后证明了方程系统存在唯一解,并给出了解的具体表达式;最后给出了误差分析,并通过数值算例表明方法的有效性.结果表明,相对于导数振荡极小化方法和应变能极小化方法,所提出的导数振荡和应变能极小化方法同时兼顾了平面分段三次参数Hermite插值曲线的保形性和光顺性.     

耗散对两个守恒律系统的影响

其中 A≥0,p'(v)<0,常出现于各种各样的应用数学问题中.例如一维不定常等熵流,非线性粘弹流中定常剪流的稳定性等问题.我们的问题是:方程(1.1)的柯西问题是否有大范围光滑解.Nishida 对这一问题进行了研究,他的结果表明,当初值条件     

Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G ² continuous contact with the control polygon at two endpoints and is G ² continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.     

有理曲线的多项式逼近

利用曲线摄动的思想给出了用多项式曲线逼近有理曲线的一种新方法．其基本步骤是对有理曲线的控制顶点进行摄动，使之产生一多项式曲线，并使摄动误差在某种范数意义之下达到最小．同时，通过适当控制摄动曲线的顶点，使逼近多项式曲线与有理曲线在两端点保持一定的连续性．这一结果可以与细分（ｓｕｂｄｉｖｉｓｉｏｎ）技术结合给出有理曲线的整体光滑的分片多项式逼近．实例表明，在某些情况下本文中的方法要优于传统的Ｈｅｒｍｉｔｅ插值方法及Ｔ．Ｗ．Ｓｅｄｅｒｂｅｒｇ和Ｍ．Ｋａｋｉｍｏｔｏ（１９９１）提出的杂交曲线逼近算法．     

SH波对界面含有半圆形脱胶的圆柱形弹性夹杂的散射的近似分析

采用Green函数法、复变函数法研究了SH波对界面附近含有半圆形脱胶的圆柱形弹性夹杂的散射,并给出了动应力集中系数的数值结果．首先,界面将整个空间分成上下两部分．在下半空间,给出在含有半圆形凸起的圆柱形弹性夹杂的弹性半空间中,水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移函数．其次,取该位移函数作为Green函数．上下空间连接时在界面处满足连续性条件,构造出半圆形脱胶裂纹,进而求出应力和位移的表达式．最后作为算例,给出了动应力集中系数的数值结果,分析了介质参数和入射波参数对动应力集中的影响情况．     

在常规故障条件下具有易损坏储备部件可修复系统的指数稳定性

研究了在常规故障条件下具有易损坏储备部件可修复系统.运用Banach空间上的线性算子半群理论,通过分析系统本质谱界经过扰动后的变化,表明在一定的条件下,系统动态解以指数形式收敛于系统的稳态解.     

一类具有储备部件的可修复人机系统的指数稳定性

研究了具有储备部件的可修复人机系统.运用Banach空间上的线性算子半群理论,证明了严格占优本征值的存在性,并通过分析系统本质谱界经过扰动后的变化,进一步表明在一定的条件下,系统动态解以指数形式收敛于系统的稳态解.     

Application of Nelder-Mead simplex method for unconfined seepage problems

In unconfined seepage problems, the phreatic line resulted from mesh deforming methods is rarely a smooth and continuous curve. The main problem is at the meeting point of the phreatic line with the down stream face of the dam where the phreatic line must be tangent to the seepage face according to the fluid continuity principle. In this paper a mesh deforming finite element method based on Nelder-Mead simplex optimization is presented to solve this problem. The phreatic line is approximated by a 4th degree polynomial and Nelder-Mead simplex method is used to calculate the polynomial’s coefficients minimizing an error function which is introduced based on the conditions on the phreatic line. Tangentiality of the phreatic line to the seepage face is introduced in the solution by a constraint in optimization procedure. The results of the presented method are verified by the results of the nonlinear finite element and other mesh deforming methods.     

Regularity for Certain Nonlinear Parabolic Systems

Abstract

By means of an inequality of Poincaré type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that a solution to certain degenerate parabolic system is locally Hölder continuous. The system is a generalization of p-Laplacian system. Using a difference quotient method and Moser type iteration it is then proved that the gradient of a solution is locally bounded. Finally using the iteration and scaling it is shown that the gradient of the solution satisfies a Campanato type integral inequality and is locally Hölder continuous.     

An Exact Solution Method for Reliability Optimization in Complex Systems

Systems reliability plays an important role in systems design, operation and management. Systems reliability can be improved by adding redundant components or increasing the reliability levels of subsystems. Determination of the optimal amount of redundancy and reliability levels among various subsystems under limited resource constraints leads to a mixed-integer nonlinear programming problem. The continuous relaxation of this problem in a complex system is a nonconvex nonseparable optimization problem with certain monotone properties. In this paper, we propose a convexification method to solve this class of continuous relaxation problems. Combined with a branch-and-bound method, our solution scheme provides an efficient way to find an exact optimal solution to integer reliability optimization in complex systems. This research was partially supported by the Research Grants Council of Hong Kong, grants CUHK4056/98E, CUHK4214/01E and 2050252, and the National Natural Science Foundation of China under Grants 79970107 and 10271073.     

连续介质中杆件系统的振动问题

弹性连续介质中杆件系统的振动问题,是工程中经常遇到的,这是弹性动力学和结构动力学的混合求解问题。按照一般的方法进行求解,似乎有很多困难,且很复杂。本文采用Lagrange乘子法,给出了这种类型平面问题的广义泛函。并通过实例说明本文方法的应用。     

Dynamic programming and the Hamilton-Jacobi method of classical mechanics

The conventional dynamic programming method for analytically solving a variational problem requires the determination of a particular solution, the optimal value function or return function, of the fundamental partial differential equation. Associated with it is another function, the optimal policy function. At each point, this function yields the value of the slope of the optimal curve to that point (or from that point, depending on the method of solution). The optimal curve itself can then be found by integration. In this paper, dynamic programming concepts and principles are used to develop two alternatives to the conventional method of solution. In the first method, a particular solution of two simultaneous partial differential equations is used to generate optimal curves by differentiations and solution of simultaneous equations. In the second method, any solution of the fundamental equation containing an appropriate number of arbitrary constants is sought. It is shown how such a function yields directly, by differentiations and solution of simultaneous equations, the optimal curve for a given problem. While the derivations to follow are new, the results are equivalent to those of a method due to Hamilton and its modification due to Jacobi.This work was completed by the author during the time of his association with the RAND Corporation Santa Monica, California.     

On Majda' Model For Dynamic Combustion

Majda's model of dynamic combustion, consists of the system,

In this paper the Cauchy problem is considered. A weak entropy solution for this system is defined, existence, uniqueness and continuous dependence on initial data are proved, as well as finite propagation speed, for initial data in . The existence is proved via the "vanishing viscosity method". Furthermore it is proved that the solution to the Riemann problem converges as to the Z–N–D traveling wave solution. In the appendices, a second order numerical scheme for the model is described, and some numerical results are presented.     

An analytical solution for sectional estimation of stress and displacement fields around a shallow tunnel

In this paper, we present a solution for the sectional estimation of the stress and displacement fields of a liner and a geomaterial near a shallow buried tunnel. The complex variable method and the discrete Fourier transform are employed in the analytical derivation. The initial boundary conditions are the nonexistence of external loads acting on the liner interior and a Fourier curve obtained by fitting the displacement along the liner interior. Stress and displacement are continuous along the excavation line. In the verification, the stress and displacement fields within the excavation line obtained by the analytical solution agree well with the numerical results. Accuracy decreases as the spatial distance between the geomaterial and the excavation line increases, and the 5% error zones are defined. The accuracy of the solution would fall greatly out of these error zones. In another word, the analytical solution is not a general solution for the whole liner–geomaterial entity, but could serve as a quick sectional estimation, and is suitable for a variety of external loads on the ground surface.     

Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case*

In this article a numerical solution for the evolution equation of a continuous time non-homogeneous semi-Markov process (NHSMP) is obtained using a quadrature method. The paper, after a short introduction to continuous time NHSMP, presents the numerical solution of the process evolution equation with a general quadrature method. Furthermore, the paper gives results that justify this approach, proving that the numerical solution tends to the evolution equation of the continuous time NHSMP. Moreover, the formulae related to some specific quadrature methods are given and a method for obtaining the discrete time NHSMP by applying a very particular quadrature formula for the discretization is shown. In this way the relation between the continuous and discrete time NHSMP is proved. Then, the problem of obtaining the continuous time NHSMP from the discrete one is considered. This problem is solved showing that the discrete process converges in law to the continuous one if the discretized time interval tends to zero. In addition, the discrete time NHSMP in matrix form is presented, and the fact that the solution to this process always exists is proved. Finally, an algorithm for solving the discrete time NHSMP is given. To illustrate the use of this algorithm for a discrete NHSMP, an example in the area of finance is presented.