非线性守恒律高阶谱粘性法的收敛性

讨论守恒型方程周期边界问题的高阶谱粘性方法逼近解的收敛性.在逼近解一致有界的假设下,通过建立其高阶导数的上界估计,证明了高阶谱粘性方法逼近解具有同二阶谱粘性方法逼近解相类似的高频衰减性质.以此为基础,用补偿列紧法证明了高阶谱粘性方法逼近解收敛于守恒型方程的物理解.     

反对称偏对称矩阵反问题的最小二乘解

该文研究了反对称偏对称矩阵反问题的最小二乘解,得到了该问题解的表达式以及该问题有解的充分必要条件.证明了其最佳逼近解的存在性和唯一性,建立了其最佳逼近解的表达式,并给出了求最佳逼近解的数值算法和算例.     

Hermitian自反矩阵反问题的最小二乘解

本文研究了Hermitian自反矩阵反问题的最小二乘解及其最佳逼近.利用矩阵的奇异值分解理论,获得了最小二乘解的表达式.同时对于最小二乘解的解集合,得到了最佳逼近解.     

广义次对称矩阵的左右逆特征对问题

本文研究广义次对称矩阵的左右逆特征对问题及其最佳逼近问题.利用广义次对称矩阵的特殊性质得到问题有解的充要条件以及通解表达式.同时给出其唯一的最佳逼近解以及求最佳逼近解的算法与实例.     

线性流形上反对称正交对称矩阵反问题的最小二乘解

该文讨论了线性流形上矩阵方程AX=B反对称正交对称反问题的最小二乘解及其最佳逼近问题. 给出了最小二乘问题解集合的表达式, 得到了给定矩阵的最佳逼近问题的解, 最后给出计算任意矩阵的最佳逼近解的数值方法及算例.     

非线性色散耗散波动方程双线性元的高精度分析

针对一类非线性色散耗散波动方程研究了双线性元逼近.基于该元的高精度分析和插值后处理技巧,对于半离散格式,在精确解的合理正则性假设下得到了H~11模意义下最优误差估计及超逼近性和超收敛结果.同时,通过构造一个新的外推格式,导出了具有三阶精度的外推解.最后,建立了一个全离散逼近格式及研究其解的超逼近性.     

一类对称正交反对称矩阵反问题的最佳逼近

讨论了一类对称正交反对称反问题的最佳逼近.利用对称正交反对称矩阵的特殊性质,给出了矩阵方程AX=B有对称正交反对称解的充要条件以及解的一般表达式;证明最佳逼近解的存在惟一性并给出其表达式;最后给出计算任意矩阵的最佳逼近解的数值方法及算例.     

线性流形上广义次对称矩阵的左右逆特征值问题

研究线性流形上广义次对称矩阵的左右逆特征值问题及其最佳逼近问题.利用广义次对称矩阵的性质及矩阵的奇异值分解得到问题的通解表达式.同时,给出其有唯一的最佳逼近解以及求最佳逼近解的算法.     

随机规划ε-逼近最优解集的Hausdorff收敛性

本文研究了随机规划ε-逼近最优解集的Haudorff收敛性条件,证明了随机规划逼近最优值的收敛性,并利用此结果给出了随机规划ε-逼近最优解集Haudorff收敛的一个充分条件.     

关于Lax-Milgram定理和Céa定理的一个注记

本文研究抽象变分问题(不必要求具有强制性)的Galerhin方法,利用泛函分析理论证明了:若变分问题的Galerkin逼近问题存在唯一解,那么它本身的解存在唯一且可由Galerhin逼近解无限逼近的充要条件是其Galerkin逼近格式具有某种稳定性.此结果是对Lax-Milgram定理和C啨a定理的补充,可以应用于不必具有强制性的变分问题.     

具变指数退化四阶抛物方程弱解的存在性

考虑一类具变指数退化四阶抛物方程的初边值问题.在一些初值的假定下,基于时间离散化方法构造逼近解,通过对逼近解的一致性估计,证明了弱解的存在性.     

向量优化问题的近似解研究

向量优化是数学规划领域中十分重要的研究方向之一,其相关基础理论与基本方法的研究具有非常重要的理论意义与应用价值.近年来,关于近似解的定义及其性质研究已成为向量优化理论与方法研究的热点.现主要介绍国内学者,特别是我们团队在向量优化问题的各类近似解和统一解概念及其发展和各类近似解与统一解的性质研究方面取得的一些重要进展.最后,提出了与向量优化问题的近似解与统一解相关的一些公开问题.     

Existence and Uniqueness of Weak Solutions to the $p$-Biharmonic Parabolic Equation

We consider an initial-boundary value problem for a $p$-biharmonic parabolic equation. Under some assumptions on the initial value, we construct approximate solutions by the discrete-time method. By means of uniform estimates on solutions of the time-difference equations, we establish the existence of weak solutions, and also discuss the uniqueness.     

A modified quasi-boundary value method for ill-posed problems

In this paper, we study a final value problem for first order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

Uniform Hölder continuity of approximate solutions to parabolic systems and its application

By means of a minimizing scheme, we constructively define approximate solutions to parabolic systems. On the basis of Campanato theory, these approximate solutions are shown to be locally Hölder equi-continuous with respect to an approximation parameter. As an application, we obtain a weak solution to an initial-boundary value problem with the same Hölder continuity.     

Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping

An algorithm to find explicit approximate solutions of an initial and terminal value problem for the forced Duffing equation with non-viscous damping is accomplished via a generalized quasilinearization method. In fact, we obtain a monotone sequence of approximate solutions converging uniformly and quadratically to a unique solution of the problem.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method

In this paper, reproducing kernel theorem is employed to solve anti-periodic solutions for Rayleigh-type equations. A simple algorithm is given to obtain the approximate solutions of the equations. By comparing the approximate solution with the exact analytical solution, we find that the simple algorithm is of good accuracy and it can be also applied to some ordinary or partial differential equations with initial-boundary value conditions and nonlocal boundary value conditions.     

轴对称渗流中自由边界问题的近似解法

１．问题的提法 本文讨论各向异性非均匀介质的轴对称稳定渗流问题,我们延用［７］的记号。设有一水井（或油井）,其截面如图１所示,ｚ轴为对称轴,Ｄ为渗流区域,其边界为ＡＢＣＥＦＡ,Ｋ＝｛ｋｉｊ（ｒ,ｚ,ｈ,ｑ）｝为对称渗流张量,它依赖于柱坐标中的ｒ,ｚ,压头ｈ＝ｚ＋ｐ／ρｇ及渗流速率其中ｐ为点（ｒ,ｚ）处的压力,ρ为流体密度,ｇ为重力加速度．ｒ０为井的半径,Ｈ１为液面的高度,同时假设当ｒ≥Ｒ时,其渗流速度Ｖ＝０． 由渗流理论,有引入热函数,流函数 Ｄ中一点）,则满足下列一阶非线性方程组其中,若（ｉ＝ １…