边界伸缩原理

文[5]提出边界伸缩原理及边界伸缩法.本文补充叙述了边界伸缩原理,并根据已有研究成果给予了较严格地证明,进一步完善了边界伸缩法理论基础.     

边界伸缩法

本文提出边界伸缩原理,并在此基础上提出边界伸缩法.不但较好地解决了边界元方法中求解边界附近区域包括边界上解的问题,而且可以方便地利用迭代过程改善求解精度.计算实例表明.本文提出的方法是十分有效的.     

利用边界元法求解一类重调和方程

边界元法(BEM)和多重互易法(MRM)相结合求解一类重调和方程.通过重调和基本解序列给出的MRM-方法和BEM, 推导出该类问题的MRM-边界变分方程, 用边界元法求解该变分方程, 从而得到重调和方程的近似解, 并给出了解的存在唯一性证明.通过数值算例说明了MRM-方法具有收敛速度快、计算精度高, 易编程等优点, 为使用边界元法数值求解重调和方程提供了方法和理论依据.适合于工程中的实际运算.     

边界曲线积分方程的小波解法

The boundary measure method is applied to transfer the form of the integral equation in order to use the collocation method or Galerkin method. A simple way to computer the coefficients of the wavelet series is also introduced. The way presented in this paper can be used to solve PDE problem in the two dimension region with any form of boundary.     

无限样条边界元及其在结构—地基耦合中的应用

本基于三维弹性力学问题的Kelvin解在无穷远处的生态，构造了一种无限样条边界元，用半空间地表圆域受均压下的解析解，验证了这类单元良好的计算精度;以它与三维样条边界元相结合，分析了弹性半空间地基板。结果表明，这种模型在分析结构与地基耦合问题上不仅具有普适性而且自由度少，计算精度高。     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

POISSON方程新的边界积分方程

POISSON方程边界值问题边界元法所应用的边界积分方程,其类型,关于未知位势导数是第一类积分方程,关于未知位势是第二类积分方程。本本文从格林公式出发,通过建立位势的单、双场守恒积分公式,推导出POISSON方程新的边界积分方程,其类型与经典方程相反,关于未知位势是第一类积分方程,关于未知位势导数是第二类积分方程。     

r=q时美式期权最佳实施边界在到期日附近的渐近展开

利用美式期权的性质及最佳实施边界S(t)满足的非线性积分方程得到S(t)的先验估计,然后利用此先验估计将对S(t)的渐近展开转化为满足方程VE(S,t)=K-S的S(t)的渐近展开,最后得到利率r与红利率q相等时美式期权最佳实施边界在到期日附近的渐近展开．     

Markov链的瞬时态与Martin边界的个数

在本文中,我们证明了含有限瞬时态的Markov链的瞬时态的个数不超过它的Martin积极流出边界的个数。     

粘弹性薄板动力响应的边界元方法（Ⅱ）——理论分析

本文中，对（１）中提出的粘弹性结构动力响应的近似边界元方法给出了必要的理论分析，得到了近似解的存在唯一性定理和误差估计。基于这些结论给出了网格宽度与基本解中截断项数的选取原则。本文中得到的理论结果和（１）中数值实验结果是一致的。     

New exact solutions to some variable coefficients problems

With the aid of computer symbolic computation system such as Maple, an extended tanh method is applied to determine the exact solutions for some nonlinear problems with variable coefficients. Several new soliton solutions and periodic solutions can be obtained if we taking paraments properly in these solutions. The employed methods are straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.     

A Note on a Functional Differential Equation with State Dependent Argument

This paper is concerned with solutions of a functional differential equation.Using Krasnoselskii's fixed point theorem,the solutions can be obtained from periodic solutions of a companion equation.     

On measure solutions of backward stochastic differential equations

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions.     

Series solutions of a semilinear wave equation

We are concerned with the reconstruction of series solutions of a semilinear wave equation with a quadratic nonlinearity. The solution which may blow up in finite time is sought as a sum of exponential functions and is shown to be a classical one. The constructed solutions can be used to benchmark numerical methods used to approximate solutions of nonlinear equations.     

The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations

The differential transform method is one of the approximate methods which can be easily applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. In this paper, we present the definition and operation of the one-dimensional differential transform and investigate the particular exact solutions of system of ordinary differential equations that usually arise in mathematical biology by a one-dimensional differential transform method. The numerical results of the present method are presented and compared with the exact solutions that are calculated by the Laplace transform method.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. I: General solutions and fundamental solutions

Four steady-state general solutions are derived in this paper for the two-dimensional equation of isotropic thermoelastic materials. Using the differential operator theory, three general solutions can be derived and expressed in terms of one function, which satisfies a six-order partial differential equation. By virtue of the Almansi’s theorem, the three general solutions can be transferred to three general solutions which are expressed in terms of two harmonic functions, respectively. At last, a integrate general solution expressed in three harmonic functions is obtained by superposing the obtained two general solutions. The proposed general solution is very simple in form and can be used easily in certain boundary problems. As two examples, the fundamental solutions for both a line heat source in the interior of infinite plane and a line heat source on the surface of semi-infinite plane are presented by virtue of the obtained general solutions.     

Exact solutions to the double Sine-Gordon equation

The double Sine-Gordon equation (DSG) with arbitrary constant coefficients is studied by F-expansion method, which can be thought of as an over-all generalization of the Jacobi elliptic function expansion since F here stands for every one of the Jacobi elliptic functions (even other functions). We first derive three kinds of the generic solutions of the DSG as well as the generic solutions of the Sine-Gordon equation (SG), then in terms of Appendix A, many exact periodic wave solutions, solitary wave solutions and trigonometric function solutions of the DSG are separated from its generic solutions. The corresponding results of the SG, which is a special case of the DSG, can also be obtained.     

Efficient numerical methods for initial-value solid-state laser problems

The difficulties of solving initial-value solid-state laser problems numerically arise from both stiffness of the problems and near-to-zero nonnegative exact solutions. Stability and non-negativity must be maintained simultaneously in the numerical solutions. Backward differentiation formulas (BDFs) is capable of dealing with stiff problems ,but is of small oscillation when time-step is large. Therefore unfortunately BDFs suffers from severe time-step restriction . In this paper,we present an optimized numerical approach, with which 3-dimensional laser problems can be solved faster and much more efficiently. These techniques can not only be used for solid-state laser systems, but can also be applied to solve other stiff problems which have near-to-zero nonnegative exact solutions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)