Yukawa型系数函数的泛函方程及其与Green函数方程的等价性

本文导出系数函数的泛函方程,找出系数函数与Green函数的泛函关系,利用泛函方法证明了Yukawa型系数函数方程与Green函数方程的等价性。     

X-椭圆方程弱解Hlder连续性的Green函数法

应用线性X-椭圆算子的Green函数与次Laplace算子基本解的局部比较原理,建立了具有有界可测系数散度型X-椭圆方程弱解的局部H(o|¨)lder连续性.以Green函数为核函数,通过holefilling技巧得到弱解满足Morrey引理条件,从而建立正则性结果,这在某种意义下取代了经典的DeGiorgi-Moser-Nash迭代技术.     

X-椭圆方程弱解H(o)lder连续性的Green函数法

应用线性X-椭圆算子的Green函数与次Laplace算子基本解的局部比较原理,建立了具有有界可测系数散度型X-椭圆方程弱解的局部H(o)lder连续性.以Green函数为核函数,通过holefilling技巧得到弱解满足Morrey引理条件,从而建立正则性结果,这在某种意义下取代了经典的De Giorgi-Moser-Nash迭代技术.     

Helmholtz方程周期Green函数及其偏导数截断误差收敛阶的分析

在应用边界元方法求解Helmholtz方程周期边值问题时,需要构造以周期Green函数或其偏导数为核函数的积分算子形式的解.由于Helmholtz方程的周期Green函数G~P是一个函数项级数,该级数的通项是Hankel函数,在数值求解中,需要对其进行截断,从而很有必要研究其截断误差.本文根据Hankel函数在变量趋于无穷大时的渐近展开式,并结合Abel不等式,证明了G~P及其一阶偏导和二阶混合偏导一致收敛,且其截断误差收敛阶均为O(1/p~(1/2)).最后,通过数值实验验证了理论证明的正确性.本文的证明方法也可被用于证明其它一些方程周期Green函数的收敛性问题.     

相对论裸粒子Green函数和碰撞积分

从相对论非平衡态统计系综理论出发,利用Bogoliubov假设和极化近似,用裸粒子Green函数方法求解相对论两体关联方程,导出用裸粒子Green函数表示的相对论动力方程碰撞积分的一般公式,导出适用于相对论等离子体的的“广义Boltzmann方程”.     

Winkler地基上固支薄板自由振动问题的准Green函数方法

将准Green函数方法应用于求解Winkler地基上固支薄板的自由振动问题.即利用问题的基本解和边界方程构造一个准Green函数,这个函数满足了问题的齐次边界条件.采用Green公式,将Winkler地基上固支薄板自由振动问题的振型控制微分方程化为第二类Fredholm积分方程.通过边界方程的适当选择,积分方程核的奇异性被克服了.数值算例表明,该方法具有较高的精度,是一种有效的数学方法.     

构造一维抛物型方程初边值问题近似解的一个注记

抛物型方程初边值问题近似解(强)的构造,[1]曾于1976年讨论过,其作法是构造一族线性无关的函数序列,并取其有限个的线性组合作为其近似解,而线性组合中的系数通过求某一线性代数方程组而获得.本文试图从构造 Green 函数出发,构造问题     

关于广义KdV方程的周期行波解的Green函数法

我们利用Green函数法来讨论一类广义KdV方程的周期行波解。通过具体作出Green函数把此方程的行波解的周期边值问题化为等价的具有对称核的积分方程。然后利用这类积分算子在周期函数空间中的紧性导出了我们需要的存在性结果。     

四阶差分方程周期边值问题的Green函数

本文研究了四阶差分方程周期边值问题的Green函数.得到了一些新的结果,推广了A. Cabada和N. Dimitrov论文中一些结果.     

简支梯形底扁球壳自由振动问题的准Green函数方法

以简支梯形底扁球壳的自由振动问题为例,详细阐明了准Green函数方法的思想.即利用问题的基本解和边界方程构造一个准Green函数,此函数满足了问题的齐次边界条件,采用Green公式,将简支梯形底扁球壳自由振动问题的振形控制微分方程化为两个耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性.最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率.数值结果表明,该方法具有较高的精度.     

Uniform Asymptotic Approximations to the Solutions of the Orr-Sommerfeld Equation Part 2. The General Theory

This paper deals with the derivation of “first approximations” to the solutions of the Orr-Sommerfeld equation which are uniformly valid in a full neighborhood of the critical point. To this order the theory is remarkably simple. The essential elements in the theory are all well known from the older heuristic theories but its general structure is substantially different. The uniform approximations are also vastly simpler than the composite approximations obtained recently by the method of matched asymptotic expansions.     

0rr-Sommerfeld方程的特征值问题及展开定理

Orr-Sommerfeld方程的特征值问题及展开定理,对于研究流体层流稳定问题具有重要意义,因此,有不少作者研究过.但据文献[6],这一问题还不能认为已完全解决.因此在本文中,我们进一步研究了这一问题,得到的具体结果是:展式的系数满足一个Paley-Wiener型不等式,它是通常完备正交系展式系数所满足的Bessel等式的一个推广:而且证明了展式是绝对一致收敛的.     

Composite Approximations to the Solutions of the Orr-Sommerfeld Equation

The method of matched asymptotic expansions is used to derive composite approximations to the solutions of the Orr-Sommerfeld equation which satisfy Olver's completeness requirement. It is shown that the inner expansions can be obtained to all orders in terms of a certain class of generalized Airy functions, and these expansions are then used to derive approximations to the connection formulae. Because of the linearity of the problem it is possible and convenient to fix the normalization of the inner and outer expansions separately and then to relate them through the central matching coefficients. The Stokes multipliers can then be expressed in terms of the central matching coefficients and the coefficients which appear in the connection formulae. Once the inner and outer expansions have been matched they can be combined, if desired, to form composite approximations of either the additive or multiplicative type. For example, the ‘modified’ viscous solutions of Tollmien emerge in a natural way as first-order composite approximations obtained by multiplicative composition; similarly, the form of the ‘viscous correction’ to the singular inviscid solutions which I conjectured some years ago emerges as a first-order additive composite approximation. Because of the completeness requirement, however, these composite approximations are valid only in certain wedge shaped domains; approximations which are valid in the complementary sectors can then be obtained by the use of the connections formulae. The theory thus provides a relatively simple and explicit method of obtaining higher approximations, and its structure permits a direct comparison of the present results not only with the older heuristic theories but also with the comparison equation method.     

流体层流运动稳定性中的Ляпунов方法

本文是[1]的继续.在本文中,利用[1]的结果我们证明了,对于流体的层流运动稳定性而言,在线性化问题中,按特征值定义与按扰动能量定义二者是完全等价的,从外,借助于Ляпунов方法,我们又证明了,如果线性化问题是渐近稳定的,当考虑非线性影响时,只要扰动能量足够小,则仍然是渐近稳定的.     

如何不用复数讲解常系数线性微分方程

在高等数学课程中,复指数函数及其导数知识的严格讲解,通常要比微分方程知识的讲解晚很多.这使得微分方程的教学在逻辑上有些不足.用复值函数解的复系数线性组合推导出实值函数解,在教学实践中,学生经常感到迷惑.不以复数的任何知识作为前提,给出了常系数微分方程的一种自然的讲解方法.     

Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming

In this paper, we present a BFGS method for solving a KKT system in mathematical programming, based on a nonsmooth equation reformulation of the KKT system. We split successively the nonsmooth equation into equivalent equations with a particular structure. Based on the splitting, we develop a BFGS method in which the subproblems are systems of linear equations with symmetric and positive-definite coefficient matrices. A suitable line search is introduced under which the generated iterates exhibit an approximate norm descent property. The method is well defined and, under suitable conditions, converges to a KKT point globally and superlinearly without any convexity assumption on the problem.     

一类三阶非线性微分方程解的不稳定性*

文献[1]讨论了非线性缓变系统的渐近稳定性,文献[2]讨论了三阶变系数线性微分方程解的不稳定性。本文应用文献[1]、[2]的方法讨论一类三阶非线性微分方程解的不稳定性。     

扰动的Couette－Poiseuille流的特征值的计算

本文考虑的问题是二维粘性渠流。对0到2000之间的雷诺数,计算了平稳扰动的Couette-Poiseuille流的下游特征值,其特征方程类似于Orr-Sommerfeld方程。所用的方法是谱方法和初值方法(复合矩阵方法).就几种有趣的流量,给出了相应的特征值的计算结果。这些特征值确定了扰动的衰减率。     

关于矩阵方程求解的另一点注记

对任意给定的矩阵A∈Pm×n,B∈Pm×s(s≤n),探讨了矩阵方程AX=B有列满秩解,同时BY=A有行满秩解的充分必要条件,并且给出了基于矩阵的等价、齐次方程组的同解、向量组的等价及线性空间语言的推广.     

Vector Eigenfunction Expansions for Plane Channel Flows

The full nonlinear initial-boundary value problem for the evolution of disturbances in plane Poiseuille flow is considered. The problem is formulated in vector form using the normal velocity and normal vorticity as components. The solution is presented as an expansion in linear eigenmodes. These modes consist of both Orr-Sommerfeld modes and modes of the normal vorticity (Squire) equation. The case of degenerating eigenmodes is also considered and it is shown that the Benney-Gustavsson normal velocity-normal vorticity resonance is a special case of a degeneracy between the vector eigenmodes. The solution to the nonlinear problem is presented as an expansion in the linear eigenmodes as well as in modes of the self-adjoint part of the linear equation. The full nonlinear solution is further reduced to small systems of coupled amplitude equations using the center manifold theorem.