边界条件齐次化辅助函数的统一形式

用分离变量法求解数理方程混合问题时,要求其第一、二、三类边界条件必须是齐次的.若为非齐次的,必须寻求恰当的辅助函数w(x,t),进行变换将其化为齐次的.本文从稳定条件下的线性非齐次边界条件出发,给出了w(x,t)的统一形式,进而将其推广到非稳定条件下的非齐次边界条件,得到w(x,t)的一般的结果.     

非齐次边界条件泛定方程的代换选择

对具有非齐次边界条件的泛定方程齐次化过程中代换的选择进行研究和探讨.基于一些相关结论和齐次化的定义得出新的研究成果,即给出对三类非齐次边界条件齐次化都适用的代换W(x,t)=A(t)x<'3>+B(t).     

化波动方程的非齐次边界为齐次边界

在一维波动方程初边值问题中,通过构造辅助函数的方法,分类将各种非齐次边界条件转化为齐次边界条件.     

椭圆重复齐次化问题W_0~(1,p)空间的收敛性

该文研究了形如-div(A(x/ε,x/ε~2)▽u_ε)=f(x)的椭圆重复齐次化问题解的收敛性,得到了Dirichlet边界条件下解在W_0~(1,p)空间的收敛率.证明所用的技巧是基于得到算子格林函数的估计.     

求解数理方程中边界条件齐次化的某些技巧

<正> 用分离变量法求解数理方程,须先将边界条件齐次化。即将问题的解分解为两个,其中一个满足非齐次边界条件,另一个满足齐次边界条件,再利用线性方程的叠加原理,则可得到原问题的解。具体地讲.就是要构造一个函数ω,使它满足非齐次边界条件。文[1]讨论了将边界条件齐次化的一般方法。但显然这样的ω不是唯     

利用齐次化原理求解常系数非齐次线性方程初值问题

文章给出利用齐次化原理求解n阶常系数非齐次线性方程初值问题的方法.通过基本问题可得到原方程的解,避免了利用常数变易法求解的诸多不便,同时也将非齐次项的形式拓展到了所有可积函数.     

非齐次偏微分方程混合问题的形式解

通过待定函数法给出了非齐次偏微分方程混合问题的形式解，从数学角度说明了齐次化原理．     

准齐次核的Hardy-Hilbert型级数不等式

定义了一类准齐次函数, 将齐次函数进行了推广. 讨论了具有准齐次核的Hardy-Hilbert型级数不等式, 并在一定条件下研究了最佳常数因子.     

非齐次偏微分方程混合问题的形式解

通过待定函数法给出了非齐次偏微分方程混合问题的形式解，从数学角度说明了齐次化原理。     

线性定常系统非齐次两点边值问题的扩展精细积分方法

提出了一种求解非齐次线性两点边值问题的高精度和高稳定的扩展精细积分方法（EPIM）.首先引入了区段量（即区段矩阵和区段向量）来离散非齐次线性微分方程,建立了非齐次两点边值问题基于区段量的求解框架.在该框架下,不同区段的区段量可以并行计算,整体代数方程组的集成不依赖于边界条件.然后引入区段响应矩阵来处理两点边值问题的非齐次项,导出了多项式函数、指数函数、正/余弦函数及其组合函数形式的非齐次项对应的区段响应矩阵的加法定理,结合增量存储技术提出了EPIM.对具有上述函数形式的非齐次项,该方法可以得到计算机上的精确解,一般形式的非齐次项则利用上述函数近似求解.最后通过两个具有刚性特征的数值算例验证了该方法的高精度和高稳定性.     

Global existence for parabolic systems by Lyapunov functions

We present a result on the global existence of classical solutions for quasilinear parabolic systems with homogeneous Dirichlet boundary conditions in bounded domains with a smooth boundary. Our method relies on the use of Lyapunov functions.     

Tangential Boundary Behaviour of Harmonic Functions on Trees

We study the behaviour of harmonic functions on a homogeneous tree from the point of view of the tangential boundary covergence.     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

An effective boundary element method for inhomogeneous partial differential equations

A method for removing the domain or volume integral arising in boundary integral formulations for linear inhomogeneous partial differential equations is presented. The technique removes the integral by considering a particular solution to the homogeneous partial differential equation which approximates the inhomogeneity in terms of radial basis functions. The remainder of the solution will then satisfy a homogeneous partial differential equation and hence lead to an integral equation with only boundary contributions. Some results for the inhomogeneous Poisson equation and for linear elastostatics with known body forces are presented.     

The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

We consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.      

Linear Elliptic Boundary Value Problems with Non‐Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non‐smooth data L^∞‐coefficients, sets with Lipschitz boundary, regular sets, non‐homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev‐Campanato spaces of functions and functionals, respectively.     

基于状态方程矩形层合板多种边界条件下的解析解

以边界位移函数方法为基础,推导了矩形层合板多种边界条件下的非齐次状态方程和定解条件.将非齐次状态方程增维齐次化,可避免积分时可能出现的数值病态问题,并简化了计算过程.边界位移沿厚度方向非线性分布假设可以适当减少数值结果收敛要求的薄层数.数值结果可作为其它数值法或半解析法的标准解.该文的方法可为分析更加复杂的边界条件问题提供参考.     

On the Plancherel Formula for the (Discrete) Laplacian in a Weyl Chamber with Repulsive Boundary Conditions at the Walls

It is known from early work of Gaudin that the quantum system of n Bosonic particles on the line with a pairwise delta-potential interaction admits a natural generalization in terms of the root systems of simple Lie algebras. The corresponding quantum eigenvalue problem amounts to that of a Laplacian in a convex cone, the Weyl chamber, with linear homogeneous boundary conditions at the walls. In this paper we study a discretization of this eigenvalue problem, which is characterized by a discrete Laplacian on the dominant cone of the weight lattice endowed with suitable linear homogeneous conditions at the boundary. The eigenfunctions of this discrete model are computed by the Bethe Ansatz method. The orthogonality and completeness of the resulting Bethe wave functions (i.e., the Plancherel formula) turn out to follow from an elementary computation performed by Macdonald in his study of the zonal spherical functions on p-adic simple Lie groups. Through a continuum limit, the Plancherel formula for the ordinary Laplacian in the Weyl chamber with linear homogeneous boundary conditions is recovered. Throughout this paper we restrict ourselves to the case of repulsive boundary conditions. Communicated by Rafael D. Benguriasubmitted 27/05/03, accepted 14/10/03     

A Reflection Principle and an Orthogonal Decomposition Concernig Hypoelliptic Equations

A jump relation for a boundary integral representation of solutions of hypoelliptic equations is described by a reflection principle. An orthogonal decomposition of L² can be proved by the jump relation. In the orthogonal complement of the space of regular functions, i.e. the space of solutions of the homogeneous equation, the inhomogeneous adjoint equation has a solution with homogeneous boundary values. As a conclusion, one obtains Sobolev's regularity theorem. Furthermore it will be proved that the existence of the orthogonal decomposition and Sobolev's regularity theorem are equivalent. Theorems of Runge's type will be proved in order to determine countable dense subsets of the space of regular functions.