微分代数方程去奇异化分析

利用去奇异化方法讨论了拟线性微分代数方程在奇点邻域内光滑解的性质.通过尺度参数的微分同胚变换,将拟线性微分代数方程转化为相应的常微分方程,从而构造出在孤立奇点邻域内的初始微分代数方程的光滑解,给出解存在的充分条件,并进一步讨论了解的性质.     

一类拟线性边值问题的激波解

本文研究了一类拟线性边值问题的激波解，在适当的条件下，利用微分不等式理论，讨论了原边值问题激波解的存在性和渐近性态。     

一类二次保形拟插值函数的研究

通过讨论一种保形拟插值的基函数与二次规范B-样条函数之间的关系,提出了一类二次保形拟插值样条函数,得到了这类保形拟插值函数在具有线性再生性质,并保持原有数据点列的单调性和凸性时分别应满足的条件,并给出几个应用实例.     

确定微分一代数方程系统指数的一个过程

一般非线性微分——代数方程系统可以写成F(y′,y,t)=0(1)其中 F、y′、y 都是维数相同的向量,在某种意义上,最简单的是线性、常系数微分——代数方程系统Ay′+By=g(t),(2)这系统可通过矩阵束(A,B)的 Kronecker 标准形得到完全了解.Gantmacher〔1〕对一     

一类拟微分算子的拟局部性质

拟微分算子理论中,算子象征决定了算子性质.本文在含参数Sobolev空间H;(τ≥τ0＞0)的框架下,讨论了相应的含参数象征的拟微分算子的有关性质,并应用微局部分析方法获得该类算子的拟局部性质.从而对经典拟微分算子作了一类推广.     

具线性增长系数的拟线性挠射问题弱解的正则性

讨论了具特殊主部和线性增长系数的n维拟线性抛物挠射问题,利用估计和平均函数方法,证明了弱解在内边界附近的一些正则性质.把这些正则性结果从线性问题推广到这种拟线性问题.     

波形松弛算法及其在计算流体力学中的应用

本文介绍求解线性常系数微分代数方程组的波形松弛算法, 基于Laplace积分变换得到该算法新的收敛理论. 进一步将波形松弛算法应用于求解非定常Stokes方程, 介绍并讨论了连续时间波形松弛算法CABSOR算法和离散时间波形松弛算法DABSOR算法.     

一类拟线性Robin问题的激波解

研究了一类Robin两点边值问题. 在适当的假设下, 利用伸长变量, 在区间内点附近构造问题解的激波层校正项. 再利用微分不等式理论, 证明了原边值问题解的存在性、一致有效性和渐近性态.     

求解延迟微分代数方程的多步Runge-Kutta方法的渐近稳定性

延迟微分代数方程(DDAEs)广泛出现于科学与工程应用领域．本文将多步Runge-Kutta方法应用于求解线性常系数延迟微分代数方程，讨论了该方法的渐近稳定性．数值试验表明该方法对求解DDAEs是有效的．     

二阶奇摄动拟线性微分系统的可解性

本文研究了一类二阶拟线件微分系统.利用边界层校正法,Green公式以及微分不等式理论,得到了一类一二阶拟线性微分系统两点边值问题的可解性条件.     

A normal form for pure differential algebraic systems

In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes-Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability.     

Non-stiff integrators for differential–algebraic systems of index 2

Non-stiff differential-algebraic equations (DAEs) can be solved efficiently by partitioned methods that combine well-known non-stiff integrators from ODE theory with an implicit method to handle the algebraic part of the system. In the present paper we consider partitioned one-step and partitioned multi-step methods for index-2 DAEs in Hessenberg form and the application of these methods to constrained mechanical systems. The methods are presented from a unified point of view. The comparison of various classes of methods is completed by numerical tests for benchmark problems from the literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal Control of Differential–Algebraic Equations of Higher Index, Part 1: First-Order Approximations

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs.     

THE EXISTENCE AND UNIQUENESS OF THE SOLUTIONS OF DAES

1Intr0ducti0nDifferential-algebraicequations(DAEs)areveryusefu1inwidefields(cf.[1]).Bydifferential-algebraicequations,wemeanthoseequati0nswhosepartsof"derivative"cann0tbeexpressedexplicitly.Forexample,weconsidertheimplicitdifferentialequationwithmappingFsm00thssufficient1y.Itisusuallyreferredt0adifferential-algebraicequation(DAE)whentherank0fD.F(t,x,p)islessthann,wheretheremightbesomepurea1gebraic,whichwecallc0nstraintequations.TheDAEs,inparticular,theexistenceanduniquenessofitssolutions…     

Stability Issues in Regular and Noncritical Singular DAEs

This paper addresses equilibrium stability issues in both regular and singular differential-algebraic equations (DAEs). We present a survey of available results and discuss some commonly-used methods in the qualitative analysis of low-index autonomous systems. Additionally, we extend the use of matrix pencil theory to the stability study of singular problems, pointing out some interesting relations between regular and singular DAEs. This framework is applied to the qualitative analysis of singular equations arising in the context of the Singularity Induced Bifurcation theorem, and also to the stability study of stationary equilibria in singular DAEs.     

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.     

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

Distributional solution theory for linear DAEs

A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Singular Values Related to DAEs in Kronecker Canonical Form

The index and the structural properties of differential algebraic equations (DAEs) are often determined by rank considerations of the derivative array. Since the Kronecker canonical form is a well-understood standard form that permits deep insight into the properties of DAEs, in this contribution we undertake an analysis of the singular values of this specific derivative array. To this end, the special structure of the obtained block matrices is pointed out, such that some formulas for the computation and estimation of eigenvalues and singular values can be applied. Actually, we explore the relationship between the spectra of particular block tridiagonal matrices and some perturbed Jacobi matrices.