A Green function for the equilateral triangle

Using the image method, a diffusional Green function has been derived analytically for a triangle, when its three sides, all of equal length, have zero intensity. Because many image sites are involved, computational assistance is employed both to construct the Green function and to use it in solving practical problems. An alternative procedure based on eigenfunctions is also summarized.Received: November 18, 2003; revised: April 6, 2004     

A fast multipole method for the evaluation of elastostatic fields in a half-space with zero normal stress

In this paper, we present a fast multipole method (FMM) for the half-space Green’s function in a homogeneous elastic half-space subject to zero normal stress, for which an explicit solution was given by Mindlin (Physics 7, 195–202 1936). The image structure of this Green’s function is unbounded, so that standard outgoing representations are not easily available. We introduce two such representations here, one involving an expansion in plane waves and one involving a modified multipole expansion. Both play a role in the FMM implementation.     

几何对称法在求取某些区域格林函数中的应用

格林函数法是数学物理方程中一种常用的方法,适用于求解狄利克雷问题.针对几种特殊区域上的上狄利克雷问题,采用几何对称法求取这些区域对应的格林函数,该方法对于该区域上格林函数的求解是有效的.     

Regions of positivity for polyharmonic Green functions in arbitrary domains

The Green function for the biharmonic operator on bounded domains with zero Dirichlet boundary conditions is in general not of fixed sign. However, by extending an idea of Z. Nehari, we are able to identify regions of positivity for Green functions of polyharmonic operators. In particular, the biharmonic Green function is considered in all space dimensions. As a consequence we see that the negative part of any such Green function is somehow small compared with the singular positive part.

     

An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign

The conjecture named after Boggio and Hadamard that a biharmonic Green function on convex domains is of fixed sign is known to be false. One might ask what happens for the triharmonic Green function on convex domains. On disks and balls it is known to be positive. We will show that also this Green function is not positive on some eccentric ellipse.     

On the oscillation property of Green’s function of a fourth-order discontinuous boundary-value problem

The paper deals with conditions under which the Green function of a multipoint boundary-value problem for fourth-order equations describing small strains of a rod fastened to a solid elastic basement and additionally fixed by “concentrated” elastic supports at separate points has the oscillation property. It is shown that the condition that the Green function is positive is necessary and sufficient for the Green function to have the oscillation property.     

Tensor Green Function for the System of Maxwell's Equations in a Layered Medium

A general method is considered for the construction of the tensor Green function for Maxwell's equations in a layered medium. An efficient algorithm for the evaluation of the tensor Green function is proposed. The properties of various components of the Green tensor are investigated.     

含有积分、反周期和带P-Laplacian算子的分数阶微分方程边值问题解的存在性

利用格林函数的性质和Banach压缩映射原理讨论了含P-Laplacian算子反周期边值问题的解.首先,求出与该边值问题相关的格林函数并给出了格林函数的性质;然后将边值问题转化为与其等价的积分方程,利用格林函数的性质及Banach压缩映射原理得到边值问题解的唯一性;最后给出实例验证结果的合理性.     

Non-singular Green’s functions for the unbounded Poisson equation in one,two and three dimensions

In this paper, we derive the non-singular Green’s functions for the unbounded Poisson equation in one, two and three dimensions using a spectral cut-off function approach to impose a minimum length scale in the homogeneous solution. The resulting non-singular Green’s functions are relevant to applications which are restricted to a minimum resolved length scale (e.g. a mesh size $h$) and thus cannot handle the singular Green’s function of the continuous Poisson equation. We furthermore derive the gradient vector of the non-singular Green’s function, as this is useful in applications where the Poisson equation represents potential functions of a vector field.     

THE COMPARISON OF GREEN FUNCTION FOR QUASI-LINEAR ELLIPTIC EQUATION

A generalization of the usual Green function to a kind of nonlinear elliptic equation of divergence form is discussed. The regularity and comparison principle of Green function in the sense of distribution are shown.     

On the Green function in visco‐elastic media obeying a frequency power‐law

In this work, we present an explicit expression for the Green function in a visco‐elastic medium. We choose Szabo and Wu's frequency power law model to describe the visco‐elastic properties and derive a generalized visco‐elastic wave equation. We express the ideal Green function (without any viscous effect) in terms of the viscous Green function using an attenuation operator. By means of an approximation of the ideal Green function, we address the problem of reconstructing a small anomaly in a visco‐elastic medium from wavefield measurements. Copyright © 2011 John Wiley & Sons, Ltd.     

Green functions for the Dirac operator under local boundary conditions and applications

In this article, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the MIT bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in some cases, solves the Yamabe problem on manifolds with boundary. Finally, using the MIT Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.     

Non-linearity of the pluricomplex Green function

We consider the pluricomplex Green function with multiple poles as introduced by Lelong. We give a partial solution to a question concerning the set where the multipole Green function coincides with the sum of the corresponding single pole Green functions.

     

不对称裂缝单井渗流模型的Green函数构造方法北大核心CSCD

不对称裂缝渗流规律可借助Green函数方法进行求解.根据基本渗流理论,建立了不对称裂缝点源数学模型,采用无因次化与Laplace变换,得到了Laplace空间的无因次点源数学模型微分方程.将未知Green函数与点源微分方程相结合,并考虑点源微分方程的齐次条件以及点源微分方程的特征,给出了如何构造Green函数使之满足点源微分方程齐次边界以及未知目标函数求解的一般方法.根据空间Green函数的对称性和连续性,得出了不对称裂缝点源模型Green函数的形式.最后通过不对称裂缝压裂直井渗流数学模型,验证了该文给出的Green函数两种形式与文献和商业试井分析软件Saphir的数值计算结果一致.     

A shortwave source near a smooth,convex hypersurface and the spectral function of the Laplace operator on a Riemannian manifold

The Tauberian theorem of B. M. Levitan reduces the question of the asymptotics of the spectral function of the Laplace operator on a smooth Riemannian manifold with boundary to the problem of constructing the asymptotics of a Green function possessing certain additional properties. The paper is devoted to the construction of the appropriate Green function for the case of a geodesically concave boundary.     

集中力作用下两相饱和介质二维位移场Green函数

由于工程场地的对称性,集中力作用下的位移场Green函数在土力学、地震工程学和动力基础方面的应用需以二维模型出现．在理论推导上Green函数的二维模型要比三维模型复杂．根据丁伯阳等人已得到的三维位移场中集中力作用下两相饱和介质位移场Green函数,采用De Hoop与Manolis给出的沿x3方向在无穷域积分方法,得到了集中力作用下两相饱和介质二维位移场Green函数．相比已有的工作,所得结果不仅简单,且是解析解．     

The space-time fractional diffusion equation with Caputo derivatives

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.     

Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient

The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional open sets if the drift function is in an appropriate Kato class.     

Green’s functions for discrete mTH-order problems*

We investigate an mth-order discrete problem with additional conditions, described by m linearly independent linear functionals. We find the solution to this problem and present a formula and the existence condition of Green??s function if the general solution of a homogeneous equation is known. We obtain a relation between Green??s functions of two nonhomogeneous problems. It allows us to find Green??s function for the same equation, but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.     

奇异四点边值问题的正解

应用不动点指数方法,在与相应线性算子第一特征值有关的条件下,得到一类奇异四点边值问题正解的存在性结果,本质地推广和改进了已有文献中的主要结论.特别地,给出了边值问题Green函数的精确表达式.