发展型偏微分方程间断有限元方法的超收敛性 献给林群教授80华诞

本文简要回顾发展型偏微分方程间断有限元方法超收敛性的若干研究工作,包括林群先生在有限元方法超收敛性质研究方向取得的部分研究成果,并对近年来相关的研究进展进行综述,展望拟开展的研究工作.     

数学天元基金管理试行办法

数学天元基金(简称天元基金)是国家为支持数学在二十一世纪率先赶上世界先进水平而设立的数学专项基金,由国家自然科学基金委员会(简称科学基金委员会)负责管理。 一、使用范围 本项基金和国家自然科学基金(简称科学基金)中的数学经费统一用于对数学的支持,彼此配合相辅相成。其主要部分用于研究工作的资助,另一部分用于改善研究工作的条件和加快优秀青年人才的培养。包括:     

复杂系统可靠性分析中的若干统计问题与进展

概述作者近十年来在复杂系统可靠性分析方法研究方面所做的部分研究工作,其中包括:复杂系统可靠性综合评估方法研究;复杂系统贮存可靠性综合评估方法研究;复杂系统可靠性综合验证方法研究;复杂数据类型下的数据填充算法等内容     

教研创新 教师专业发展的需要和必然

在历次课堂和教学改革中,数学学科一直走在行动的前列.在当下举国共建创新型国家的坚定步伐中,中国教育学会中学数学教学专业委员会为了进一步指导和规范中学数学教研工作,组织专门力量正在研制中学数学教研工作指导意见.部分省(市)教研部门也就教研创新开展了相关研究工作.例如,安徽省教育科学研究院徐子华老师主持的安徽省教育规划课题《(中学数学)教研创新与教师专业成长有效结合的研究》(编号:JG09010)已经开展了一系列研究活动.笔者有幸参加了以上两个项目的部分研究工作.下面,我就中学数学教研工作的现状、教研创新的必要性、教研创新的主要形式,以及教研创新需要注意的问题等方面作一些分析和思考,以求教于大家.     

“控制室科研工作介绍”专辑

<正>前言2012年是中国科学院系统控制重点实验室的前身中国科学院数学所控制理论研究室成立50周年.大约在一年以前,系统控制室的一些科研团队成员建议在系统科学研究所30周年所庆材料中系统控制重点实验室科研工作简介的基础上,由《系统科学与数学》出一个专辑比较系统地介绍系统控制室的研究工作,以便于进行学术交流.本专辑所收集的文章对系统控制室的研究人员在系统控制相关领域的部分代表性研究工作做了总结,其中包括自校正调节器、变界截尾随机逼近算法、矩阵半张量积的理论与应     

企业开展现场统计研究的几个问题

本文通过调查 ,对企业开展现场统计研究工作存在的三个问题 ,即认识问题、条件问题、环境问题 ,进行了阐述与分析 ;提出了促进企业开展现场统计研究工作的四点建议     

计算机辅助几何设计中极小曲面造型的研究进展

极小曲面在工程领域有着广泛应用,因此将其引入计算机辅助几何设计领域具有重要意义.详细概述了近年来计算机辅助几何设计领域中极小曲面造型的研究工作,按照造型方法的不同,可将现有工作分为精确造型方法和逼近造型方法两类.精确造型方法主要包括两个部分:某些特殊极小曲面的控制网格表示与构造;等温参数多项式极小曲面的挖掘与性质.逼近造型方法主要包括3个部分t基于数值计算的逼近方法;基于线性偏微分方程的逼近方法;基于能量函数最优化的逼近方法.最后对这些方法进行了分析比较,并讨论了极小曲面造型中有待进一步解决的问题.     

离散大系统关于部分变元的关联稳定性

一、问题的提出在文献[1]中,曾提出了关于部分变元的稳定性问题.1957年,首先引进函数 V(t,x_1,…,x_n)关于变元 x_1,…,x_m(m≤n)为定号函数等概念,并给出了系统关于部分变元稳定性和渐近稳定性的定理.随后,许多学者从事这方面的研究工作.实际上这是与这个问题本身具有很重要的实用性分不开的.意义下的稳定性是针对系统的所有状态变量而言的.但实际上很多稳定性问题并不要求(或不可能要求)用同样的方式处理所有的状态变量.到目前为止,虽然关于全变元稳定性的许多结果已平行地推广到部分变元稳定性问题上来,但是从具体问题出发,来解决     

非线性系统运动稳定性的一种判据

对于自治的非线性系统来说,只要其线性部分系数矩阵的特征值不属于临界情形,其无扰运动在其足够小的邻域内的稳定性完全可以由其线性部分的特征值确定.关于线性系统的稳定性,已有不少简单易行的判别方法,而关于非线性系统的稳定性,很多数学家和力学家作了大量的研究工作;但大都是针对特殊类型的非线性系统解决了一些问题,直到现在为止,还没有普遍适用于任何的非线性系统的简单易行的判别方法.本文所给的是判别非线性系统稳定性的充要条件,常用的克拉索夫斯基方法只是这一方法的一个特例[1],[2].     

重物落水后运动过程的数学建模

分析了影响重物落水运动过程的主要因素,根据各因素权重关系及相关物理理论建立了多因素融合的大实心方砖落水过程模型,对重物在水中的运动分成部分浸入水中和重物完全浸入水体两个阶段.定性描述了两连接物体落水后耦合运动过程,并与实测数据进行拟合,根据对比的结果,实现了误差分析,根据误差原因优化模型以提高其准确度和适用性,并对未来的试验和研究工作提出了一些建议,以提高试验数据的质量.     

Asymptotic Behaviour and Global Existence of Solutions for Some Classes of Nonlinear Parabolic Equations

The paper deals with the asymptotic behaviour and global existence of solutions for some classes of nonlinear parabolic equations in regard to the monotone properties of the nonlinear term. The asymptotic behaviour of the solutions of initial-boundary value problem for nonlinear parabolic equations is studied via the method of differential inequalities in order to obtain oscillation criterion for the solutions. Existence of extremal solutions of semilinear elliptic and parabolic equations is investigated via monotone iterative methods. The extremal solutions are obtained via monotone iterates.     

Separation of linear components in nonlinear boundary heat control

By means of an additional substitution a parabolic control problem with some nonlinear boundary condition will be decoupled into some control problem with linear parabolic state equations and an appropriate nonlinear mapping. This separation allows the use of efficient techniques e.g. Fourier methods, to determine the solution of linear parabolic state equations. Essential properties of the mapping used in the transformation are studied. Further, the application of piecewise constant discretizations of the controls in connection with the proposed splitting is discussed.     

High order nonlinear parabolic equations

The basic results and methods of the theory of high order nonlinear parabolic equations are described. In the first chapter boundary problems for quasilinear parabolic equations having divergent form are considered. In the second chapter nonlinear parabolic equations of general form are considered. Attention is mainly paid to methods of study of nonlinear parabolic problems. In particular, the methods of monotonicity and compactness, the method of a priori estimates, the functional-analytic method, etc. are described.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 89–166, 1990.     

具有未知源的某些非线性伪抛物型方程的反问题

考 虑具有未 知源项的 某些非 线性伪 抛物 型方程 的反演 问题． 首先 将伪抛 物型 方程初 边值问 题化为非线 性发展方 程 Ｃｏｕｃｈ ｙ 问题,然 后,利用半 群理论,论 证发展 方程反问 题解的存 在唯一 性,最后, 利用不 动点方法得到 伪抛物型方程反 问题的可解性     

Blow up for nonlinear parabolic equations with time degeneracy under dynamical boundary conditions

We study the blow up behaviour of nonlinear parabolic equations including a time degeneracy, under dynamical boundary conditions. For some exponential and polynomial degeneracies, we develop some energy methods and some spectral comparison techniques and derive upper bounds for the blow up times.     

Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $\rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.     

Stabilizing Semilinear Parabolic Equations

In this paper we prove the internal feedback stabilization of steady-state solutions of semilinear parabolic equations and introduce a controller synthesis methodology based on finite element approximations of the original PDEs. Numerical tests are given for some one- and two-dimensional nonlinear parabolic equations.     

Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions

The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator–prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived.     

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions. Received December 21, 2000; accepted August 30, 2001.     

ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS

This paper discusses the accelerating of nonlinear parabolic equations. Two iterative methods for solving the implicit scheme new nonlinear iterative methods named by the implicit-explicit quasi-Newton （IEQN） method and the derivative free implicit-explicit quasi-Newton （DFIEQN） method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iteration method can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov （JFNK） method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear （inner） iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iteration methods are presented in confirmation of the theory and comparison of the performance of these methods.