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.     

On Mappings of Plane Domains by Solutions of Second-Order Elliptic Equations

In this paper, we study sufficient conditions for the one-to-one solvability of secondorder partial differential equations in a plane Jordan domain. For a continuous one-to-one and orientation-keeping map of the boundary of a Jordan domain to the rectifiable boundary of some other Jordan domain, we prove the following property: If the Cauchy integral whose measure is generated by this map is bounded by some constant in the exterior domain, then the solution to the corresponding Dirichlet problem in the domain with this boundary function maps these domains one-to-one. In the proof of the main result we use integral representations of equation solutions, particularly, properties of Fredholm-type integral equations on the domain boundary.     

Duality theorems in some overdetermined boundary value problems

We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.     

A System of Plane Elasticity Canonical Integral Equations and Its Application

In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,     

圆外平面弹性问题的边界积分公式

将边界上的应力函数及其法向导数展开为罗朗级数,与复应力函数的罗朗级数的表达式对比,可以确定罗朗级数的各系数,再利用傅利叶级数和卷积的几个公式进行计算,得到应力函数边界积分公式．通过边界的应力函数及其法向导数的积分,直接得到圆外应力函数值,并给出几个算例,表明结果用于求解单位圆外平面弹性问题十分方便．     

The boundary integral method for the linearized rotating Navier-Stokes equations in exterior domain

In this paper, we apply the boundary integral method to the linearized rotating Navier-Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the linearized rotating Navier-Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence and uniqueness of solution. Finally, we study the finite element approximation for the coupled problem and obtain the error estimate between the solution of the coupled problem and its approximation solution.     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

WEIGHTED KOPPELMAN－LERAY－NORGUET FORMULAS ON A LOCAL q－CONCAVE WEDGE IN A COMPLEX MANIFOLD

A weighted Koppelman-Leray-Norguet formula of (r, s) differential forms on a local q-concave wedge in a complex manifold is obtained. By constructing the new weighted kernels, the authors give a new weighted Koppelman-Leray-Norguet formula with-out boundary integral of (r, s) differential forms, which is different from the classical one. The new weighted formula is especially suitable for the case of the local q-concave wedge with a non-smooth boundary, so one can avoid complex estimates of boundary integrals and the density of integral may be not defined on the boundary but only in the domain. Moreover, the weighted integral formulas have much freedom in applications such as in the intervolation of functions.     

The boundary integral method for the steady rotating Navier–Stokes equations in exterior domain (I): the existence of solution

In this paper, we apply the boundary integral method to the steady rotating Navier–Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and a infinite domain, we obtain a coupled problem by the steady rotating Navier–Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence of solution in a convex set.     

局部扰动半平面上的散射问题

该文讨论半平面上有局部扰动情况下的散射问题．通过位势理论,应用边界积分方程的方法研究了该问题解的存在与唯一性．主要方法是运用对称反射,使该无界区域上的散射问题变成一个有界区域上的散射问题,只是这一有界区域的边界不光滑．通过仔细分析相应的边界积分算子,作者得到了其解的存在与唯一性．     

Shape-explicit constants for some boundary integral operators

Among the well-known constants in the theory of boundary integral equations are the coercivity constants of the single-layer potential and the hypersingular boundary integral operator, and the contraction constant of the double-layer potential. Whereas there have been rigorous studies how these constants depend on the size and aspect ratio of the underlying domain, only little is known on their dependency on the shape of the boundary. In this article, we consider the homogeneous Laplace equation and derive explicit estimates for the above-mentioned constants in three dimensions. Using an alternative trace norm, we make the dependency explicit in two geometric parameters, the so-called Jones parameter and the constant in Poincaré's inequality. The latter one can be tracked back to the constant in an isoperimetric inequality. There are many domains with quite irregular boundaries, where these parameters stay bounded. Our results provide a new tool in the analysis of numerical methods for boundary integral equations and in particular for boundary element based domain decomposition methods.     

复流形上具有非光滑边界的强拟凸域上带权因子的Koppelman-Leray公式

利用权因子,我们得到了复流形上边界不必光滑的强拟凸域上（狆,狇）微分形式的带权因子的Ｋｏｐｐｅｌｍａｎ Ｌｅｒａｙ公式及其 方程的带权因子的解,其特点是不含有边界积分,从而避免了边界积分的复杂估计．其次,引进了权因子,带权因子的积分公式在应用上具有更大的灵活性．     

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.     

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。     

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．     

The norm and the essential norm of the double layer elastic and hydrodynamic potentials in the space of continuous functions

We study a class of matrix integral operators which appear as limit values of the double layer potentials. We find general representations for the norms and for the essential norms of such operators in the space of continuous vector-valued functions. These representations are specified for boundary integral operators of linear isotropic elasticity theory and hydrodynamics of viscous incompressible fluid under the assumption that there is an angle point on the boundary of a plane domain and a conic point or an edge on the boundary of a three-dimensional domain.     

Hybrid boundary element formulation for acoustic wave propagation

The so‐called hybrid stress boundary element method (HSBEM) is introduced in a frequency domain formulation for the computation of acoustic radiation and scattering in closed and in finite domains. Different from other boundary element formulations, the HSBEM is based on an extended Hellinger‐Reissner variational principle and leads to a Hermitian, frequency‐dependent stiffness equation. Due to this, the method is very well suited for treating fluid structure interaction problems since the effort for the coupling the structure, discretized by a finite elements, and the fluid, discretized by the HSBEM is strongly reduced. To arrive at a boundary integral formulation, the field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions weighed by source strength. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. Comparing to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix for exterior domain problems are communicated here.     

Analysis of united boundary‐domain integro‐differential and integral equations for a mixed BVP with variable coefficient

The mixed (Dirichlet–Neumann) boundary‐value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary‐domain integro‐differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential‐type operators defined on open sub‐manifolds of the boundary and acting on the trace and/or co‐normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary‐domain integro‐differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.