调和方程自然边界元Shannon 小波方法

1 引言 调和方程无论是在数学上还是在物理学中都占有重要地位,它有很多不同的物理背景,在力学和物理学中研究的许多问题都可归结为调和方程的边值问题,所以对调和方程进行深入研究有重要意义.余德浩教授在[3]中主要对调和方程在典型域(即单位圆,上半平面)上的情形进行了考虑.特别地,对单位圆的情形给出了刚度矩阵系数的计算公式和调和方程解的存在唯一性.本文采用由冯康教授[1]开创的自然边界元方法和Galerkin小波方法相耦合,对上半平面的调和方程Neumann问题进行了研究,得到十分有效的计算结果.     

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。     

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.     

Clifford分析中广义超正则函数的一个非线性边值问题

讨论了Cliffrd分析中广义超正则函数的一个非线性边值问题.首先将广义超正则函数分解为两个奇异积分算子,然后给出了广义超正则函数的Plemelj公式及相关奇异积分算子的性质,最后利用Schauder不动点原理证明了广义超正则函数的一个非线性边值问题的解的存在性及积分表达式.     

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.     

椭圆外区域上Helmholtz问题的自然边界元法

本文研究椭圆外区域上Helmholtz方程边值问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式及自然积分方程,给出了自然积分方程的数值方法.由于计算的需要,我们详细地讨论了Mathieu函数的计算方法(当0相似文献   

抛物型初边值问题的边界积分-微分方程及其边界元方法

本文提出和研究了抛物型方程Neumann初边值问题的一个新的边界归化方法。它将原始初边值问题归化成一类新的边界积分-微分方程。由此导出一种新的既保持原始问题的自伴性,又具有可积弱奇性积分核的边界变分方程和边界元方法,给出了近似解在各种范数意义下的先验误差估计。     

四元数分析中超球与双圆柱区域上的正则函数

本文讨论了四元数分析中的正则函数Ｕ（ｚ）（满足方程ｚＵ（ｚ）＝０,ｚ＝ｘ１＋ｉｘ２＋ｊｘ３－ｋｘ４）及其边值问题,给出了超球与双圆柱区域上的四元数正则函数的Ｃａｕｃｈｙ积分公式,获得了一般区域上正则函数的无穷次可微性;给出了定义在超球与双圆柱区域边界上的四元数函数可正则开拓到区域内的条件;讨论了满足非齐次方程ｚＦ＝ｆ的四元函数Ｆ（ｚ）的Ｄｉｒｉｃｈｌｅｔ和Ｎｅｕｍａｎｎ边值问题;获得了超球与双圆柱区域上这两种边值问题解的积分表示．     

双正侧函数的非线性带位移边值问题

Ｃｌｉｆｆoｒｄ分析中的双正则函数是一类广义正则函数,它的研究是近年来函数论领域内的一个热门分支,本文研究双正则函数的非线性带位移的边值问题．设计积分算子,将边值问题转化成积分方程问题,借助于积分方程理论和Ｓｃｈａｕｄｅｒ不动点理论证明了边值问题解的存在性并给出了解的积分表达式．     

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.     

Canonical Integral Equations of Stokes Problem

The canonical boundary reduction, suggested by Feng Kang, also can be applied to the bidimensional steady Stokes problem. In this paper we first give the representation formula for the solution of the Stokes problem via two complex variable functions. Then by means of complex analysis and the Fourier analysis, we find the expressions of the Poisson integral formulas and the canonical integral equations in three typical domains. From these results the canonical boundary element method for solving the Stokes problem can be developed.     

Numerical Solutions of Harmonic and Biharmonic Canonical Integral Equations in Interior or Exterior Circular Domains

Elliptic boundary-value problems can be reduced to integral equations on the boundary by many different ways. The canonical reduction, suggested by Prof. Feng Kang, is a natural and direct approach of boundary reduction. This paper gives the numerical method for solving harmonic and biharmonic canonical integral equations in interior or exterior circular domains, together with their convergence and error estimates. Using the theory of distributions, the difficulty caused by the singularities of integral kernel is overcome. Results of several numerical calculations verify the theoretical estimates.     

Canonical Boundary Element Method for Plane Elasticity Problems

In this paper, we apply the canonical boundary reduction, suggested by Feng Kang, to the plane elasticity problems, find the expressions of canonical integral equations and Poisson integral formulas in some typical domains. We also give the numerical method for solving these equations together with their convergence and error estimates. Coupling with classical finite element method, this method can be applied to other domains.     

A family of potentials for elliptic equations with one singular coefficient and their applications

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double- and simple-layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball.     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

A Fast Algorithm for Two-Dimensional Elliptic Problems

In this paper, we extend the work of Daripa et al. [14–16,7] to a larger class of elliptic problems in a variety of domains. In particular, analysis-based fast algorithms to solve inhomogeneous elliptic equations of three different types in three different two-dimensional domains are derived. Dirichlet, Neumann and mixed boundary value problems are treated in all these cases. Three different domains considered are: (i) interior of a circle, (ii) exterior of a circle, and (iii) circular annulus. Three different types of elliptic problems considered are: (i) Poisson equation, (ii) Helmholtz equation (oscillatory case), and (iii) Helmholtz equation (monotone case). These algorithms are derived from an exact formula for the solution of a large class of elliptic equations (where the coefficients of the equation do not depend on the polar angle when written in polar coordinates) based on Fourier series expansion and a one-dimensional ordinary differential equation. The performance of these algorithms is illustrated for several of these problems. Numerical results are presented.     

Generalized boundary value problems for analytic functions with convolutions and its applications

In this paper, we first establish a locality theory for the Noethericity of generalized boundary value problems on the spaces . By means of this theory, of the classical boundary value theory, and of the theory of Fourier analysis, we discuss the necessary and sufficient conditions of the solvability and obtain the general solutions and the Noether conditions for one class of generalized boundary value problems. All cases as regards the index of the coefficients in the equations are considered in detail. Moreover, we apply our theoretical results to the solvability of singular integral equations with variable coefficients. Thus, this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.     

Solution of boundary-value problems for a degenerating elliptic equation of the second kind by the method of potentials

In this paper we study basic boundary value problems for one multidimensional degenerating elliptic equation of the second kind. Using the method of potentials we prove the unique solvability of the mentioned problems. We construct a fundamental solution and obtain an integral representation for the solution to the equation. Using this representation we study properties of solutions, in particular, the principle of maximum. We state the basic boundary value problems and prove their unique solvability. We introduce potentials of single and double layers and study their properties. With the help of these potentials we reduce the boundary value problems to the Fredholm integral equations of the second kind and prove their unique solvability.     

Coupling Canonical Boundary Element Method with FEM to Solve Harmonic Problem over Cracked Domain

Using the canonical boundary reduction, suggested by Feng Kang, coupled with the finite element method, this paper gives the numerical solutions of the harmonic boundary-value problem over the domain with crack or concave angle. When the coupling is conforming, convergence and error estimates are obtained. This coupling removes the limitation of the canonical boundary reduction to some typical domains, and avoids the shortcoming of the classical finite element method, because of which the accuracy is damaged seriously and the approximate solution does not reflect the behaviour of the solution near the singularity. Numerical calculations have verified those conclusions.