二阶半线性奇摄动边值问题解的渐近行为

该文研究非法向双曲条件下的二阶半线性奇摄动边值问题解的渐近行为.利用边界层函数法,构造了区间端点处的代数型边界层,获得了问题的一致有效渐近解;利用微分不等式理论,证明了解的存在性以及渐近解与精确解之间的误差估计.通过一个典型的算例,验证了该文的理论结果.     

具有无穷边界值的二次非线性奇摄动边值问题的双边界层

研究了具有无穷边界值的二次非线性奇摄动边值问题的双边界层,利用边界层校正函数,构造其渐近解,并利用微分不等式理论,给出了一致有效渐近估计.最后给出算例验证了相关结论的正确性.     

具有内部层的奇摄动微分差分方程的渐近解

该文针对一类非线性奇摄动微分差分方程边值问题,用边界层函数法构造了一致有效的渐近解.由于偏差效因,边界层函数的确定困难很多.作者用"缝接法"不但证明了光滑解的存在性,而且给出了余项估计.     

函数不连续的二阶拟线性奇摄动边值问题

讨论了函数不连续情况下二阶拟线性奇摄动边值问题,用边界层函数法和轨道的光滑缝接,构造了问题的形式渐近解,并在整个区间上证明了形式渐近解的一致有效性,把吉洪诺夫系统中的函数光滑条件推广到了不连续情况.     

在子区间上奇异摄动的一类边值问题解的渐近分析

研究了在子区间上奇异摄动的一类半线性二阶微分方程边值问题,用边界层函数法构造出问题的形式渐近解,借助微分不等式理论证明了渐近解的一致有效性.     

带非线性无穷大边值条件的二阶半线性奇摄动问题

考虑关于带非线性无穷大边界值条件的二阶半线性奇摄动边值问题.基于边界层校正的思想,分别构造了左、右端点邻域的指数型及代数型的边界层校正函数,得到了问题的渐近解;根据微分不等式理论,获得了该问题解的存在性、渐近解的一致有效性以及渐近解的误差估计.还着重探讨了一定的稳定性条件下问题所能允许的边界值的奇异程度问题.通过一个典型的算例,验证了文中理论结果的正确性以及渐近解的高精度性.     

一类带有Neumann边界条件奇异摄动反应扩散方程的周期解

主要研究一类奇异摄动反应扩散方程周期解的存在性和渐近稳定性.首先,利用边界层函数法,构造出形式渐近解,基于微分不等式理论,得到了周期解的存在性.然后讨论周期解在李雅普诺夫意义下的渐近稳定性.最后,由具体例子说明该方法的有效性.     

高阶非线性奇摄动微分方程的三点边值问题

本文讨论了一类高阶非线性奇摄动微分方程的三点边值问题.根据小参数的不同次幂,分情况补充相应的边界条件.运用边界层函数法,构造了形式渐近解,并得到解的存在唯一性和渐近解的一致有效性.最后用数值计算结果印证了结论.     

一类分数阶微分方程初值问题的奇摄动

研究了一类非线性分数阶微分方程加权初值问题的奇异摄动.在适当的条件下,首先求出了原问题的外部解,然后利用边界层函数法构造出解的初始层项,并由此得到解的形式渐近展开式,最后利用微分不等式理论,讨论了问题解的渐近性态,得到了原问题解的一致有效的渐近估计式.     

一类拟线性奇异摄动方程组的空间对照结构

本文研究带慢变量的右边不连续的拟线性奇异摄动方程组的空间对照结构.利用边界层函数法构造了该方程组的形式渐近解,并运用"缝接法"证明问题解的存在性以及渐近解的一致有效性.最后,通过例子验证了所得结果的有效性.     

Initial value problems for singular and nonsmooth second order differential inclusions

We study the existence of W^2,1 solutions for singular and nonsmooth initial value problems of the type whereT > 0 is a priori fixed, x₀, x₁ ∈ ?, and F: [0, T ] × ? → ??(?) \ {??} is a multivalued mapping. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dupin indicatrices and families of curve congruences

We study a number of natural families of binary differential equations (BDE's) on a smooth surface in . One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE.

More generally, we consider a natural class of BDE's on such a surface , and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

     

Localized triangular differential quadrature

A localized triangular differential quadrature method is introduced in this article. Not only is the existing limitation on the approximation order in the triangular differential quadrature eliminated but also the convergent rate is enhanced in the new method. As an example to validate the new method, elastic torsion of prismatic shaft with regular polygonal cross section is studied and excellent agreement with available theoretical and analytic solutions is reached. It is believed that the present work further widens the applicability of the triangular differential quadrature technique. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 682–692, 2003     

ON THE CONVEXITY OF SOME BANACH SPACES

51. IntroductionThe convexity of Banach spaces is an important topic in functional analysis and playsan importals role in indnite dimensional holomorphy. In order to study the geometricproperties of a Banach spacet Clarksonl3] and Krein (independently) introduced the veryimportallt class of strict convex spaces. In the same paper [3], Clarkson also introduced thestronger notion of uniform convexity. Since Clarkson's paper many authors have definedand studied the classes of Banach spaces lyi…     

From an iteration formula to Poincaré's Isochronous Center Theorem for holomorphic vector fields

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincaré, for the condition under which a singularity of a holomorphic vector field is an isochronous center.

     

变系数线性齐次常微分方程组的λ-矩阵求解法

进一步讨论了微分变换矩阵的性质 ,指出了变系数线性齐次微分方程组 ,通过因变量变换化为常系数线性齐次方程组的充要条件     

微分周形式与稀疏微分结式

代数周(Chow)形式和代数结式是代数几何的基本概念,同时还是消去理论的强大工具.一个自然的想法是在微分代数几何中发展相应的周形式和结式理论.但是由于微分结构的复杂性,在本文的研究工作之前,微分结式只有部分结果,而微分周形式与稀疏微分结式理论一直没有得到发展.本文的主要结果包括:第一,发展一般(generic)情形的微分相交理论,作为应用,证明一般情形的微分维数猜想.第二,初步建立微分周形式理论.对不可约微分代数簇定义微分周形式并证明其基本性质,特别地,给出微分周形式的Poisson分解公式,引入微分代数簇的主微分次数这一不变量并证明一类微分代数闭链的周簇和周坐标的存在性.作为应用,首次严格定义微分结式,证明其基本性质.第三,初步建立稀疏微分结式理论.引入Laurent微分本性系统的概念,定义稀疏微分结式,证明其基本性质,特别地,引入微分环面簇的概念,给出稀疏微分结式阶数和次数界的估计,并基于此给出计算稀疏微分结式的单指数时间算法.     

Equivalence and symmetries of first order differential equations

In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations . That means, the transformed unknown function is obtained by means of the change of the independent variable and subsequent multiplication by a nonvanishing factor. Instead of the common direct calculations, we use some more advanced tools from differential geometry; however, the exposition is self-contained and only the most fundamental properties of differential forms are employed. We refer to analogous achievements in literature. In particular, the generalized higher symmetry problem involving a finite number of invariants of the kind is compared to similar results obtained by means of auxiliary functional equations.     

Equivalence and symmetries of second-order differential equations

In this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations = φ(x), ȳ = ȳ() = L(x) + y(x), = () = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed. This research has been conducted at the Department of Mathematics as part of the research project CEZ: Progressive reliable and durable structures, MSM 0021630519.