考虑横向剪切变形和旋转惯性的层合正交异性球壳的动态响应

本文建立了计及横向剪切变形和旋转惯性的复合材料轴对称层合圆柱正交异性球壳的运动方程．在此基础上，用有限差分法计算了球壳在轴对称动力载荷下的动态响应，并讨论了材料参数、结构参数和横向剪切变形的影响．     

钝体绕流非定常解

本文结合目前流场显示的研究课题,对方块物体和山形物体的钝体绕流在起动阶段的运动情况,进行相应的数值模拟.并用有限差分方法求解二维不可压缩流体运动的N-S方程的非定常解.对差分格式中的显式,隐式和交替方向隐式几种格式进行了讨论.最后用显式和交替方向隐式方法计算了山形物体和方块物体在起动阶段的运动情况.     

基于有限差分法的二阶双曲型分布参数系统的迭代学习控制

研究了一类分布参数系统基于有限差分法的迭代学习控制问题,该类分布参数系统由二阶双曲型偏微分方程所构建.针对系统所满足的初、边值条件,基于有限差分法构建得到迭代学习控制律,利用迭代收敛原理,证明这种学习律能使得系统的状态跟踪误差沿迭代轴方向收敛到原点的一个小邻域内.数值仿真验证了所提算法的有效性.     

有限环上典型群的阶

本文通过“取模”，“取值和”，“取积”等方法，将有单位元的有限环Ｒ上典型群阶的计算转化为有限域上典型群阶的计算，并计算了Ｒ上ｎ－维自由模Ｖ_ｎ（Ｒ）中ｋ－维自由子模的个数．     

拟线性Sobolev方程的特征有限元格式的交替方向预处理迭代解及其分析

考虑数值求解具有对流项的高维拟线性Ｓｏｂｏｌｅｖ方程,构造了特征有限元格式,提出用交替方向预处理迭代法求特征有限元格式在每一时间步所产生的代数方程组的近似解,整个计算过程仅对一个可方向交替的预处理矩阵求逆一次,大大降低了计算量．证明了迭代解的最佳L^2模误差估计,并给出了算法的拟优工作量估计．     

广义神经传播方程的A.D.I.有限元分析

广义神经传播方程是神经传播方程的更一般形式，是一类非常重要的非线性发展方程，在生物、力学诸领域有实际背景．有关其解的性质，已有许多讨论I‘－‘］，但数值计算和数值分析方面，还没有研究．而实际当中，研究模型的数值方法和定量计算，往往是非常重要的．本文着重讨论一类具有非线性边值条件的广义神经传播方程的交替方向有限元方法及其数值分析．算法上，采用交替方向有限元，化高维问题为低维问题，在缩减计算量的同时，保持高精确度．分析上，采用Sobolev投影，简化论证，得到理想的稳定性和收敛性结果．1方程模型及有限元数值…     

消除毛细管电泳槽道中弯道导致的扩散效应的新方法

首先分析了毛细管通道中流动的弯道效应及其产生的原因,接着在建立电渗流场数学模型的基础上,用有限差分法对弯道处内外壁面上不同电荷分布时的扩散进行了数值模拟．根据计算结果,提出了一种基于改变弯道处内外壁面上电荷分布的新方法,以此使流场的弯道效应最弱．同时还建立了分析和确定弯道处最佳电荷分布的优化方法．结果表明,该方法能极大地消除毛细管通道中的流动弯道效应．     

一类非线性抛物型方程非线性边值问题整体范围内差分方法的可解性

在非线性抛物型方程边值问题可解性的研究中,用有限差分法进行先验估计也是一个常用的方法。但使用有限差分法所得出的可解性往往是局部的,同时在非线性边界的估计中也遇到了一定的困难。 1962年,K.Rektorys在[1][2]中首次用有限差分法证明了一类非线性抛物型方程的边值问题在整体范围内的可解性,但他只研究了第一边值问题及一些简单的其它边值问题,对于非线性边值问题,我们还没有见到用有限差分法取得成功的报导。     

抛物型方程非齐次边值问题的推广型LOD有限差分及有限元格式

1引言本文考虑区域Ω=[0,1]~d(d=2,3)上的非齐次抛物型方程第一边值问题(?)-C_1△u C_2u=f(x,t),x∈Ω,t∈(0,T],(1．1) u(x,0)=u_0(x),x∈Ω,(1．2) u(x,t)=(?)(x,t),x∈(?)Ω,t∈(0,T],(1．3)其中C_1,C_2为常数且C_1＞0,C_2≥0．对于以上问题,可以使用有限差分方法及有限元方法进行离散,并采用交替方向方法求解．交替方向方法能够将高维问题转化为一系列的一维问题进行计算,具有计算量少,计算稳定且易于并行实现等优点,在大规模科学计算中起着非常重要的作用,一直是计算数     

带绝对值符号的函数的积分法

本文拟通过一些例子探讨带绝对值符号的函数的定积分计算的规律和方法．一、基本方法解决这类积分的基本思路是：用分段函数表示被积函数，以便去掉绝对值符号，然后利用定积分的可加性，分段进行计算．1．找“零点”，分区间，脱去绝对值符号树三计算积分，其中E为闭区间[0，4π]中使积分式有意义的一切值所成之集合．解由已知条件知找“零点”，为此解方程cosx=0在积分区间上的“零点”为此时积分鞠间分成一般地，计算积分．我们就需要求出的所有“零点”，并用这些“零点”把积分区间分为几个部分区间，然后讨论f（X）在各部分区间上的…     

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式(ADI)差分方法.向后欧拉(Euler)方法联立一阶卷积求积公式处理时间方向的离散,有限差分方法处理空间方向的离散,并进一步构造了ADI全离散差分格式.然后将二维问题延伸到三维问题,构造三维空间问题的ADI差分格式.基于离散能量法,详细证明了全离散格式的稳定性和误差分析.随后给出了2个数值算例,数值结果进一步验证了时间方向的收敛阶为一阶,空间方向的收敛阶为二阶,和理论分析结果一致.     

Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.     

裂缝孔隙介质中二相驱动问题的交替方向有限元方法及理论分析

考虑裂缝孔隙介质中二相驱动问题的数值方法及理论分析。对压力方程采用混合有限元方法,对裂缝和岩块系统上的饱和度方程采用交替方向有限元方法,证明了交替方向有限元格式具有最优Ｌ２模和Ｈ１模误差估计。     

一类单调变分不等式的非精确交替方向法

交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明.     

A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation

In this paper, we investigate the numerical solution of the three-dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first-order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three-dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non-linear term. Thereafter we prove a positive-type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.     

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A high‐order finite difference method for the two‐dimensional complex Ginzburg–Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876–899, 2015     

广义交替近似梯度算法的线性收敛分析

针对两个可分凸函数的和在线性约束下的极小化问题,在交替方向法的框架下,提出广义的交替近似梯度算法.在一定的条件下,该算法具有全局及线性收敛性.数值实验表明该算法有好的数值表现.     

A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation

In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids. Then, we use the Richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid. Finally, we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid. During this process, we use alternating direction implicit (ADI) method to solve the resulting linear systems. Numerical experiments are conducted to verify the accuracy and effectiveness of the present method.     

ALTERNATING DIRECTION FINITE ELEMENT METHOD FOR SOME REACTION DIFFUSION MODELS

This paper is concerned with some nonlinear reaction - diffusion models. To solve this kind of models, the modified Laplace finite element scheme and the alternating direction finite element scheme are established for the system of patrical differential equations. Besides, the finite difference method is utilized for the ordinary differential equation in the models. Moreover, by the theory and technique of prior estimates for the differential equations, the convergence analyses and the optimal L^2- norm error estimates are demonstrated.     

Uniformly convergent novel finite difference methods for singularly perturbed reaction–diffusion equations

In this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.