拟线性抛物组第一边值问题的差分方法(续)

本文在文献[1]的基础上,继续讨论拟线性抛物组第一边值问题的有限差分方法.首先就弱隐式差分组解的存在性以及强隐式和弱隐式差分解的唯一性进行了论证,最后给出了差分解的收敛性.特别是对显式和弱隐式差分组,给出了收敛性条件,即关于差分步长限制条件.     

双曲-双曲奇异摄动混合问题的一致收敛格式

本文构造了二阶双曲-双曲奇异摄动混合问题的差分格式,给出了差分解的能量不等式,并证明了差分解在离散范数下关于小参数一致收敛于摄动问题的解.     

Finite difference schemes for Burgers' equation

针对Burgers方程的初边值问题建立了全离散两层加权中心差分格式,得到了差分解的L2模估计,证明了差分解的存在性、收敛性和稳定性,并且得到了显格式和弱隐格式对于步长Γ和h的限制条件.     

美式期权定价问题的变网格差分方法

本文提出一种求解美式期权定价自由边值问题的变网格差分方法.通过建立一个自由边界所满足的方程,利用变网格技术可同时求出期权的差分解和最佳执行边界.本文分别讨论了显式和隐式变网格差分格式,并给出了差分解的收敛性和稳定性分析.数值实验表明本文算法是一个非常有效的期权定价算法.     

钝体分离旋涡流动的区域分解、杂交数值模拟——Ⅰ.理论方法及其应用

为克服涡旋法不能精确预计物体附近小尺度流动结构的理论缺陷,减少高Reynolds数流动N-S方程差分解的困难,本文提出一种区域分解、杂交耦合N-S方程有限差分解及涡旋法的新的数值模型和理论方法.将流场分解为内外两区,在靠近物体表面、范围为O(R)的内区进行N-S方程有限差分解,外区作Lagrange-Euler涡旋法解,建立了分区流动的联结、耦合条件,给出了杂交耦合求解的数值计算方法.用本方法作了Re=10²,10³的圆柱绕流计算,考察了区域交界面位置变化时解的稳定性.与全场N-S方程解及实验结果的比较表明本文方法能精确预计流动分离及近场流动的详细结构,并可有效地计算流动的总体特性,且比全场N-S方程解显著节省机时和计算量.     

一类奇异方程两点边值问题的差分—样条校正解

本文考虑一类奇异方程两点边值问题的差分解和样条数值解法,证明了差分解,样条解分别从两侧逼近精确解,从而得到高精度的差分-样条校正解. 考虑如下形式的奇异边值问题:     

一类奇异方程两点边值问题的差分—样条校正解

本文考虑一类奇异方程两点边值问题的差分解和样条数值解法,证明了差分解,样条解分别从两侧逼近精确解,从而得到高精度的差分-样条校正解. 考虑如下形式的奇异边值问题:     

带五次项的非线性Schrödinger方程的守恒差分格式

本文研究带有五次项的非线性Schrödinger方程初边值问题的有限差分法,其中方程中二阶偏导数项的系数、五次项的系数及初值满足下面的条件（1.6）.针对此问题,我们研究了一个守恒差分格式,在条件（1.6）下,差分解的$L^{\infty}$模先验估计被得到.在此基础上,我们得到了差分解最优$L^2$模的误差估计.     

二维粘弹性问题的广义差分法及其数值模拟

本文利用广义差分法讨论了二维粘弹性问题.建立了二维粘弹性问题的广义差分格式,给出了一种新的初始值近似,证明了广义差分解的最优L~p估计和W~1,p(2≤p≤∞)模估计,同时得到了广义的Ritz—Volterra投影和广义差分解之间的超收敛的W~1,p(2≤p≤∞)模的误差估计.最后给出了一个数值算例以验证该方法的可行性.     

线性混合模型协方差阵的谱分解的一种新方法及其应用

对平衡线性混合模型, 随机效应的设计阵具有一定结构.定义了一种新的矩阵序, 借助于这种新序, 提出了协方差阵谱分解的一种新方法.与现有的两种方法相比较, 新方法的突出的特点是能够给出协方差阵不同特征值的精确个数, 以及谱分解中不同特征值对应的投影阵与随机效应的设计阵之间的关系. 基于新的谱分解结果,(1) 证明了平衡随机模型的方差分析估计为最小方差无偏估计; (2) 证明了在一定条件下, 一般平衡线性混合模型的方差分析估计也具有最小方差无偏性; (3) 给出了一般混合模型的极大似然方程显示解存在的一个较易验证的判定定理, 并给出了显示解存在时解的一般形式; (4) 清晰地显示了谱分解估计的构造原理, 并找到了谱分解估计与方差分析估计相等的充要条件.     

On existence of solutions of differential‐difference equations

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non‐linear differential‐difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential‐difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich‐type theorem and Malmquist‐type theorem about differential equations to the case of differential‐difference equations. Copyright © 2015 John Wiley & Sons, Ltd.     

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

一阶多时滞半线性差分方程解的振动性

本文讨论了一阶多时滞半线性和中立型差分方程解的振动性，在一定的条件下，获得了其解有振动性的一些充分条件。     

Permanence in kolmogorov-type systems of delay difference equations

In this paper some general results have been established on the permanence of the two-species Kolmogorov-type autonomous difference system with time delays, and they are applied to the two-species Lotka-Volterra-type difference system with time delays     

Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.     

Numerical prediction of recirculation flows with free convection encountered in gas-agitated reactors

Flows in a gas-agitated reactor have been predicted by a finite difference procedure. The free-convection phenomena in the gas-liquid mixtures have been accounted for by the calculation of a void fraction determined from the gas flow rate. Computations have been performed for two different situations: first, with the allowance of slip between gas and liquid phases, and second, without any slip. Reasonable agreement has been achieved between the measurements.     

A compact finite difference method for the solution of the generalized Burgers–Fisher equation

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The Symmetrical Dissipative Difference Schemes for Quasi-Linear Hyperbolic Conservation Laws

In this paper we construct a new type of symmetrical dissipative difference scheme. Except discontinuity these schemes have uniformly second-order accuracy. For calculation using these, the simple-wave is very exact, the shock has high resolution, the programing is simple and the CPU time is economical. Since the paper [1] introduced that in some conditions Lax-Wendroff scheme would converge to nonphysical solution, many researchers have discussed this problem. According to preserving the monotonicity of the solution preserving monotonial schemes and TVD schemes have been introduced by Harten, et. According to property of hyperbolic wave propagation the schemes of split-coefficient matrix (SCM) and split-flux have been formed. We emphasize the dissipative difference scheme, and these schemes are dissipative on arbitrary conditions.     

一类反应扩散方程组的保持正性的差分格式

本文研究一类具有正解的反应扩散方程组的有限差分解法.构造了一个保持正性的差分格式.利用离散的最大值原理证明了差分格式解的非负性,有界性及差分格式的无条件稳定性.这些估计的证明不依赖于微分方程的解而仅仅与初边值条件有关.当微分方程的解适当光滑时,证明了差分格式的一致收敛性.最后给出了数值计算结果,并与以往方法进行了比较.计算结果说明了本文给出的方法的有效性.     

一类推广的离散分数阶Gronwall不等式

离散不等式,特别是离散的Gronwall不等式已被广泛应用于差分方程的研究.近年来,分数阶微分方程引起很多学者的关注.因此,利用一种新的分数阶和分的定义和不等式的方法,讨论一类更一般的离散分数阶Gronwall不等式.