包含—受扭迴转轴的半空间的性质*

本文研究包含有一根部份嵌入的迴转轴的半空间的性质.不用知道一给定的嵌入的轴的扭转问题的精确解,这些性质能指出此半空间的位移或应力场的某些特点并且有时可以用来检查数值解.文中给出嵌入半空间的受扭的刚性圆柱的轴的表面上的正确的应力分布的检查的例子.     

Vertical oscillations of multimass system on elastic half space with a cylindrical cavity

A problem on oscillations of a multimass system (MS) is considered on an elastic half space with a cylindrical cavity. Equations of motions of an MS are given, which are modeled by masses that are connected by springs and dampers. A motion of the half space with a cavity is characterized by a transmitting function,which is known from a solution of a contact problem with vertical oscillations of a die on the half space given. The conditions of interrelation of the MS with the base close the system of algebraic linear equations for determining amplitudes of oscillations of each element of the MS. Translated from Dinamicheskie Sistemy, No. 7, pp. 13–18, 1988.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

嵌在弹性半空间的刚性变直径圆轴的扭转*

本文用线载荷积分方程法(LLIEM)研究嵌在弹性半空间的刚性变直径圆轴的轴对称扭转.利用将“半空间的点环力偶”(PRCHS)这一虚的基本载荷沿对称轴的圆轴区间中分布就能使本问题归结为一个一维的非奇异的Ferdholm第一种积分方程,且很易用数值求解.文中给出刚性圆锥,圆柱,圆锥柱嵌在弹性半空间的扭转的数值计算的例并和他人的已知结果相比较.文中也给出了刚性半球嵌在弹性半空间的扭转的精确解.     

Waves from a Deep Source of Longitudinal Waves in an Elastic Half Space with a Free Boundary

The nonstationary propagation of waves on the surface of an elastic half space from a deep expansion source (model of an explosion in a half space) is examined. Exact solutions are obtained in the form of integrals with finite limits and the general solution is calculated. Algebraic expressions are obtained for the Rayleigh wave. The transition of Rayleigh waves at the surface of the half space is studied. Calculations of Rayleigh waves from discontinuous pulsed sources are presented.     

BOUNDARY LAYER SOLUTION FOR p-SYSTEM WITH ARTIFICIAL VISCOSITY

An initial-boundary values problem in the half space （0, ∞ ） for p-system with artificial viscosity is investigated. It is shown that there exists a boundary layer solution. It is further proved that the boundary layer solution is nonlinear stable with arbitrarily large perturbation. The proof is given by an elementary energy method.     

Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Green function are split into their half advanced + half retarded and half advanced–half retarded components, and closed form solutions for these components are then obtained in terms of a Horn function and a Kampé de Fériet function respectively. Series solutions for the Fourier coefficients are given in terms of associated Legendre functions, Bessel and Hankel functions and a hypergeometric function. These series are derived either from the closed form 2-dimensional hypergeometric solutions or from an integral representation, or from both. A simple closed form far-field solution for the general Fourier coefficient is derived from the Hankel series. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented. Fourth order ordinary differential equations for the Fourier coefficients are also given and discussed briefly.     

SH波对界面含有半圆形脱胶的圆柱形弹性夹杂的散射的近似分析

采用Green函数法、复变函数法研究了SH波对界面附近含有半圆形脱胶的圆柱形弹性夹杂的散射,并给出了动应力集中系数的数值结果．首先,界面将整个空间分成上下两部分．在下半空间,给出在含有半圆形凸起的圆柱形弹性夹杂的弹性半空间中,水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移函数．其次,取该位移函数作为Green函数．上下空间连接时在界面处满足连续性条件,构造出半圆形脱胶裂纹,进而求出应力和位移的表达式．最后作为算例,给出了动应力集中系数的数值结果,分析了介质参数和入射波参数对动应力集中的影响情况．     

Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Green function are split into their half advanced + half retarded and half advanced–half retarded components, and closed form solutions for these components are then obtained in terms of a Horn function and a Kampé de Fériet function respectively. Series solutions for the Fourier coefficients are given in terms of associated Legendre functions, Bessel and Hankel functions and a hypergeometric function. These series are derived either from the closed form 2-dimensional hypergeometric solutions or from an integral representation, or from both. A simple closed form far-field solution for the general Fourier coefficient is derived from the Hankel series. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented. Fourth order ordinary differential equations for the Fourier coefficients are also given and discussed briefly.     

一维可压Navier—Stokes方程初边值问题局部解的存在性

我们证明了半空间中一维可压Navier—Stokes方程初边值问题局部解的存在性,证明主要是利用了能量方法．     

Numerical solution of nonlinear three‐point boundary value problem on the positive half‐line

In this paper, we present an efficient numerical algorithm to solve the three‐point boundary value problem on the half‐line based on the reproducing kernel theorem. Considering the boundary conditions including a limit form, a new weighted reproducing kernel space is established to overcome the difficulty. By applying reproducing property and existence of the orthogonal basis in the weighted reproducing kernel space, the approximate solution is constructed by the orthogonal projection of the exact solution. Convergence has also been discussed. We demonstrate the accuracy of the method by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Anisotropic elastodynamics in a half space: An analytic method for polynomial data

An initial boundary value problem for the dynamic system of anisotropic elasticity in a half space is studied in the paper. A novel method of finding an exact solution of this problem for a special polynomial form of initial data and inhomogeneous term of the system is described. On the base of this method the simulation of elastic waves in different homogeneous anisotropic half spaces is implemented.     

Decay rate for the incompressible flows in half spaces

We show that the time decay rate of norm of weak solution for the Stokes equations and for the Navier–Stokes equations on the half spaces are if the initial data and for . We also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai's solution formula for the Stokes equations. It has been known up to now that the decay rate on the half space was , which was obtained by Borchers and Miyakawa [1] and Ukai [9]. Received: 3 November 1999; in final form: 10 May 2000 / Published online: 17 May 2001     

椭圆孔边裂纹对SH波的散射及其动应力强度因子

采用复变函数和Green函数方法求解具有任意有限长度的椭圆孔边上的径向裂纹对SH波的散射和裂纹尖端处的动应力强度因子．取含有半椭圆缺口的弹性半空间水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移解作为Green函数,采用裂纹“切割”方法,并根据连续条件建立起问题的定解积分方程,得到动应力强度因子的封闭解答．讨论了孔洞的存在对动应力强度因子的影响．     

On the lack of exact controllability for mild solutions in Banach spaces

It is shown that exact controllability in finite time for linear control systems given on an infinite dimensional separable Banach space in integral form (mild solution) can never arise using locally L₁-controls, if the operator through which the control acts on the system is compact. This improves a previous result of the author, by removing the assumption that the state space have a basis. It is suggested by the recent discovery that a separable Banach space need not have a basis.     

关于实方阵的几个问题

本文讨论如下内容：１．把有关对称正定（半正定）的一些性质推广到广义正定（半正定）。２．给定ｘ∈Ｒｍ×ｍ，∧为对角阵，求ＡＸ＝ｘ∧在对称半正定矩阵类中解存在的充要条件及一般形式，并讨论了对任意给定的对称正定（半正定）矩阵Ａ，在上述解的集合中求得Ａ，使得     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Decay properties of solutions to the incompressible magnetohydrodynamics equations in a half space

We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The L^r‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L¹ and L ^∞ decay rates of its first order derivatives with respect to space variables, are derived by using L^q ? L^r estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd.     

Diophantine Quadratic Equation and Smith Normal Form Using Scaled Extended Integer Abaffy–Broyden–Spedicato Algorithms

Classes of integer Abaffy–Broyden–Spedicato (ABS) methods have recently been introduced for solving linear systems of Diophantine equations. Each method provides the general integer solution of the system by computing an integer solution and an integer matrix, named Abaffian, with rows generating the integer null space of the coefficient matrix. The Smith normal form of a general rectangular integer matrix is a diagonal matrix, obtained by elementary nonsingular (unimodular) operations. Here, we present a class of algorithms for computing the Smith normal form of an integer matrix. In doing this, we propose new ideas to develop a new class of extended integer ABS algorithms generating an integer basis for the integer null space of the matrix. For the Smith normal form, having the need to solve the quadratic Diophantine equation, we present two algorithms for solving such equations. The first algorithm makes use of a special integer basis for the row space of the matrix, and the second one, with the intention of controlling the growth of intermediate results and making use of our given conjecture, is based on a recently proposed integer ABS algorithm. Finally, we report some numerical results on randomly generated test problems showing a better performance of the second algorithm in controlling the size of the solution. We also report the results obtained by our proposed algorithm on the Smith normal form and compare them with the ones obtained using Maple, observing a more balanced distribution of the intermediate components obtained by our algorithm.     

The boundary layer analysis for stokes equations on a half space

The time dependent Stokes equations on a half space are considered. We decompose the solution of these equations in an inviscid solution: a boundary layer solution and a correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction is shown to be of the same order of magnitude of the square root of the viscosity.