高维Dirichlet问题数值解的概率方法

本文用概率方法求得高维Dirichlet内问题和外问题在一般区域上的数值解\bd 高维漂移布朗族对停时具有强马氏性, 它在球面上的击中时和位置分布已知, 再利用Dirichlet问题解的随机表达式, 我们可以获得高维Dirichlet问题的数值解     

一类复蒙日—安培方程Dirichlet问题数值解探讨(一)

蒙日一安培方程是高度非线性的偏微分方程,因此求取它的数值解非常困难.本文对第一类Cartan-Hartogs域上的复蒙日-安培方程Dirichlet问题数值解进行了探讨.首先,把该问题化为一个二阶非线性常微分方程的两点边值问题的数值解.其次,在一些特殊的情况下,得到了该方程的Dirichlet问题解的显表达式,它可以用来检验该问题的数值解.     

一个带有小参数的二维椭圆方程的渐近性分析

通过数值方法研究在边界充分(逐段)光滑区域上的带有小参数的二维椭圆方程在部分Dirichlet边界控制下的渐近性问题.对于一维的情形求解析解的结果,对高维问题提出类似的问题.但高维问题解析求解一般不可能,因此采用数值分析的方法.数值结果表明,在所选的条件下,边界值对小常数仍然不是解析的.     

Dirichlet外问题的概率数值方法

针对定解区域是无界区域的Dirichlet外问题,提出了一种新的有效的概率数值方法,它是从解的随机表达式出发,将无界区域上的问题转化成区域边界上的问题.此时,只要在边界上进行剖分,将问题离散化,然后在无界区域外的有界区域内构作一个辅助球,并且利用布朗运动、漂移布朗运动从球外一点出发,首中球面的位置和时间的分布等,就可以获得Dirichlet外问题的数值解.     

Dirichlet外问题的概率数值方法

针对定解区域是无界区域的Dirichlet外问题,提出了一种新的有效的概率数值方法,它是从解的随机表达式出发,将无界区域上的问题转化成区域边界上的问题.此时,只要在边界上进行剖分,将问题离散化,然后在无界区域外的有界区域内构作一个辅助球,并且利用布朗运动、漂移布朗运动从球外一点出发,首中球面的位置和时间的分布等,就可以获得Dirichlet外问题的数值解.     

利用Block-Pulse函数求解分数阶泊松方程数值解

借助于二维Block-Pulse函数求解分数阶泊松方程的数值解,并讨论了Dirichlet边界条件,方法是基于Block-Pulse函数的定义及性质,并结合相应的分数阶微分算子矩阵将原问题转化为含有未知变量的代数方程组,进而离散未知变量,求得原问题的数值解.而且还对所提方法进行了误差分析,最后给出的数值算例也验证了所提算法的有效性及可行性.     

二维波动方程部分Dirichlet边界控制正则性的数值分析

通过数值方法,研究边界充分(逐段)光滑区域上的二维波动方程在部分Dirichlet边界控制下的正则性问题.数值结果表明:在所选条件下,系统是Salamon-Weiss意义下正则的.     

长方体内布朗运动的初出分布与 Dirichlet 问题

<正> 由于熟知的 n 维布朗运动与 Dirichlet 问题的紧密联系,对于 n 维空间中的正则开集D,给出从 D 内出发的布朗运动的初出分布,也就可给出 D 上 Dirichlet 问题的解.迄今这类分布多是对球、半空间等给出具体的表达式,它们对应于 Dirichlet 问题的已有结果(如 Poisson 公式).本文对 n 维空间中一类典型区域——n 维长方体给出布朗运动的初     

一类Monge-Ampère方程的特征值问题

对一类Monge-Amp(e)re方程的特征值问题进行了研究.通过移动平面法证明了在凸对称区域内,Dirichlet问题的C2凹(凸)解一定是对称的.进而通过对常微分方程和椭圆形偏微分方程的讨论,得到一类n维单位球上特征值问题的非平凡解的存在性和正则性结果.     

二维瞬态热传导问题的无单元Galerkin法分析

采用无单元Galerkin(element-free Galerkin,EFG)法求解具有混合边界条件的二维瞬态热传导问题.首先采用二阶向后微分公式离散热传导方程的时间变量,将该问题转化为与时间无关的混合边值问题;然后采用罚函数法处理Dirichlet边界条件,建立了二维瞬态热传导问题的无单元Galerkin法;最后基于移动最小二乘近似的误差结果,详细推导了无单元Galerkin法求解二维瞬态热传导问题的误差估计公式.给出的数值算例表明计算结果与解析解或已有数值解吻合较好,该方法具有较高的计算精度和较好的收敛性.     

Discrete Dirichlet boundary value problems with upper semicontinuous right-hand sides

This paper deals with the relationship between solutions of Dirichlet boundary value problems (BVPs) for second order systems of differential inclusions with upper semicontinuous right-hand sides and associated numerical discrete Dirichlet BVPs of second order difference inclusions. First, the existence and estimate of solutions to the discrete BVP is discussed uniformly with respect to the discrete step size. Then convergence of solutions of the numerical discrete BVP and the corresponding semicontinous BVP is studied. Related results are also mentioned which motivated our study of this problem.     

运用漂移布朗族的高维狄利克莱问题的数值解

本文对高维狄利克莱问题的数值解提出了一种新的有效的求解方法。这种方法运用了解的随机表达式、球面击中时和位置的分布以及漂移布朗族的强马氏性。     

SUPERCONVERGENCE AND NONSUPERCONVERGENCE OF THE SHORTLEY-WELLER APPROXIMATIONS FOR DIRICHLET PROBLEMS

We are concerned with the numerical solutions of Dirichlet problems of elliptic equations. The convergence behavior of numerical solutions by using Shortley-Weller approximation is considered. We give three examples and prove that they have properties of nonsuperconvergence near any boundary point, superconvergence near a side and superconvergence near a corner, respectively.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

Weak solutions to hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary conditions

In this paper, we consider the weak solutions of hyperbolic problems subject to inhomogeneous Dirichlet and Neumann boundary conditions. Using Fourier–Galerkin method, we obtain approximate solutions of the problems and test the obtained results on numerical examples by MAPLE®.     

On a Fourier Method of Embedding Domains Using an Optimal Distributed Control

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. This method can be easily extended to three-dimensional problems. The method is based on formulating the problem as an optimal distributed control problem inside a rectangle in which the arbitrary domain is embedded. A periodic solution of the equation under consideration is constructed easily by making use of Fourier series. Numerical results obtained for Dirichlet problems are presented. The numerical tests show a high accuracy of the proposed algorithm and the computed solutions are in very good agreement with the exact solutions.     

Bilateral estimates for solutions of boundary value problems in Kirchhoff plates

Dirichlet boundary value problems are studied for thin elastic plates on an elastic foundation within Kirchhoff's classical model. The aim is to construct dual problems that make it possible to obtain bilateral error estimates for approximate solutions. In the absence of an elastic foundation, the dual functionals are maximized on function sets whose elements satisfy certain differential restrictions. The theory is illustrated by means of a numerical example.     

An unified numerical approach of steady convection between two parallel plates

This paper considers the problem of laminar forced convection between two parallel plates. We present an unified numerical approach for some problems related to this case: the problem of viscous dissipation with Dirichlet and Neumann boundary conditions and the Graetz problem. The solutions of these problems are obtained by a series expansion of the complete eigenfunctions system of some Sturm-Liouville problems. The eigenfunctions and eigenvalues of this Sturm-Liouville problem are obtained by using Galerkin’s method. Numerical examples are given for viscous fluids with various Brinkman numbers.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.