Banach空间一类H-增生算子的混合拟变分包含的邻近算子方程

本文在Banach空间中引入一类H-增生算子的混合拟变分包含,并提出求该变分包含问题解的邻近点法.通过H-增生算子的预解算子技术,建立了混合拟变分包含问题与邻近算子方程的等价关系,由这个等价关系得到求解邻近算子方程的迭代算法,该算法收敛于上述混合拟变分包含问题的解.     

电动力学电磁场边值问题的广义变分原理

给出了线性各项异性电磁场边值问题的广义虚功原理表达式,运用钱伟长教授提出的方法建立了该问题的广义变分原理,可直接反映该问题的全部特征,即4个Maxwell方程、2个场强-位势方程、2个本构方程和8个边界条件．继而导出了一族有先决条件的广义变分原理．作为例证,导出了两个退化形式的广义变分原理,和已知的广义变分原理等价．此外还导出了两个修正的广义变分原理,可为该问题提供杂交有限元模型．建立的各广义变分原理可为电磁场边值问题的有限元应用提供更为完善的理论基础．     

临界群与二阶差分方程解的多重性

研究了一类二阶非线性差分方程两点边值问题解的多重性.当该问题的非线性项在无穷远点具有特殊的渐近线性性质时,利用变分方法,结合临界群与Morse理论,同时考虑正、负能量泛函的临界点,不论该问题是否发生共振,均证明了它至少存在两个非零解.     

离散共振问题的非平凡解

研究了一类二阶非线性差分方程两点边值问题非平凡解的存在性.假设该问题在无穷远点及零点处均是共振的,利用变分方法,同时考虑正、负能量泛函的临界点,在一定的假设条件下,通过临界群的计算,证明了该问题至少存在一个非平凡解.     

一类具有p-Laplacian算子的分数阶差分方程边值问题的正解

该文研究一类具有p-Laplacian算子的分数阶差分方程边值问题.借助离散型Jensen不等式,考虑该问题与相应的不带有p-Laplacian算子的分数阶差分方程边值问题之间的关系,并运用不动点指数理论获得该问题正解的存在性.     

双调和方程区域分解法的矩阵分析

1 引 言 设Ω为R~2平面上的有界凸多边形区域,边界Ω适当光滑,四阶调和方程的边值问题 △~2u=f, Ω Ⅰ)u=△u=0, Ω Ⅱ)u=u/n=0, Ω 这儿△~2表示双调和算子,f∈L_2(Ω),问题Ⅰ)为简支板的平衡方程,问题Ⅱ)为固定边界板的平衡方程。对于问题Ⅰ)、Ⅱ)的混合变分形式分别为     

二阶差分方程Robin边值问题非平凡解的存在唯一性.

借助变分方法和临界点理论,研究了二阶差分方程Robin边值问题非平凡解的存在唯一性,推广和完善了已有的一些结果.     

一类非线性所变分不等式

本文利用伪单调算子理论研究如下变分不等式问题，求ｘ∈Ｍ，使得并将所得结果应用于拟线性椭圆型边值问题的求解。     

小参数在高阶导数项的椭圆型方程第一边值问题的一致收敛差分格式

本文讨论了曲边区域上小参数ε在高阶导数项的椭圆型方程第一边值问题,从一致收敛的必要条件出发构造了特殊的差分格式,证明了差分方程问题解的一致收敛性,估计了收敛的阶数,并讨论了差分方程解的渐近性态.     

论拉普拉斯方程第三边值问题与变分问题的等价性

首先给出了柱坐标系下拉普拉斯方程的第三边值问题,进而证明了拉普拉斯方程的第三边值问题等价于一个泛函变分的极值问题,最后指出了将拉普拉斯方程第三边值问题转换为等价的泛函变分极值问题的好处.     

A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second‐order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi‐infinite strip is considered. The boundary impedance (Neumann‐to‐Dirichlet map) is computed as the square root of an operator using the standard three‐point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx‐imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.     

板弯曲问题的具两组高阶基本解序列的MRM方法

讨论了双参数地基上薄板弯曲问题.利用两组高阶基本解序列,即调和及重调和基本解序列,采用多重替换方法(MRM方法),得到了板弯曲问题的MRM边界积分方程.证明了该方程与边值问题的常规边界积分方程是一致的.因此由常规边界积分方程的误差估计即可得到板弯曲问题MRM方法的收敛性分析.此外该方法还可推广到具多组高阶基本解序列的情形.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Analytic solutions to heat transfer problems on a basis of determination of a front of heat disturbance

With the use of additional boundary conditions in integral method of heat balance, we obtain analytic solution to nonstationary problem of heat conductivity for infinite plate. Relying on determination of a front of heat disturbance, we perform a division of heat conductivity process into two stages in time. The first stage comes to the end after the front of disturbance arrives the center of the plate. At the second stage the heat exchange occurs at the whole thickness of the plate, and we introduce an additional sought-for function which characterizes the temperature change in its center. Practically the assigned exactness of solutions at both stages is provided by introduction on boundaries of a domain and on the front of heat perturbation the additional boundary conditions. Their fulfillment is equivalent to the sought-for solution in differential equation therein. We show that with the increasing of number of approximations the accuracy of fulfillment of the equation increases. Note that the usage of an integral of heat balance allows the application of the given method for solving differential equations that do not admit a separation of variables (nonlinear, with variable physical properties etc.).     

变厚度圆薄板非对称非线性弯曲问题

首先将直角坐标系中的横向变厚度薄板的大挠度方程,转化到极坐标系中的变厚度圆薄板的非对称大挠度方程·　此方程和极坐标系中径向、切向两个平衡方程联立求解·　将物理方程和中面应变非线性变形方程,代入3个平衡方程,可得用3个变形位移表示的3个非对称非线性方程·　用Fourier级数表示的解代入基本方程,获得相应的基本方程·　在周边夹紧边界条件下,用修正迭代法求解·　作为算例,研究了余弦形式载荷作用下的问题,还给出了载荷与挠度的特征曲线,曲线依据变厚度参数变化而变化,其结果和物理概念完全吻合·     

变厚度扁锥壳的非线性固有频率

借助于变厚度圆薄板非线性动力学变分方程和协调方程,给出了变厚度扁薄锥壳的非线性动力学变分方程和协调方程·　假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取变厚度扁锥壳中心最大振幅为摄动参数,采用摄动变分法,将变分方程和微分方程线性化·　对周边固定的圆底变厚度扁锥壳的非线性固有频率进行了求解;一次近似得到了变厚度扁锥壳的线性固有频率,三次近似得到了变厚度扁锥壳的非线性固有频率,且绘出了固有频率与静载荷、最大振幅、变厚度参数的特征曲线图·　为动力工程提供了有价值的参考·     

Optimal Control for non–isothermal crystallization of polymers

In this paper an optimal control problem for polymer crystallization is investigated. The crystallization is described by a non–isothermal Avrami–Kolmogorov model. The boundary temperature serves as control variable. The cost functional takes the spatial variation of the crystallinity and the final degree of crystallization into account. The resulting boundary control problem for a parabolic equation coupled with two ordinary differential equations is treated by an adjoint variable approach. A steepest descent algorithm is used to solve the problem numerically. Simulations illustrate the applicability and performance of the optimization algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

四元数分析中的T算子与两类边值问题

本文研究四元数分析中的非齐次 Ｄｉｒａｃ方程．引入了这类方程的分布解即 Ｔ算子,证明了Ｔ算子的一些性质并考察了非齐次Ｄｉｒａｃ方程的Ｄｉｒｉｃｈｌｅｔ边值问题,并将结果推广到高阶非齐次Ｄｉｒａｃ方程及这种方程的一类边值问题的情况．