一类三维拟抛物粘性扩散方程有限差分逼近的长时间行为(英文)

研究了一类带有周期边界条件的三维拟抛物粘性扩散方程有限差分解的长时间行为.证明了数值解的存在唯一性,离散系统全局吸引子的存在性,差分格式的长时间稳定性和收敛性.此外,我们给出了上半连续性.     

二维空间中半线性摄动波动方程初值问题解的渐近理论

研究二维空间中具初值问题的半线性波动方程解的渐近理论,在二次连续的古典空间中得到了形式近似解的渐近合理性在长时间范围内成立,这一结果描述了渐近解的长时间存在性.作为所得到的渐近理论的应用,对二维空间中的一个特殊波动方程作出了分析.     

边界耦合的牛顿渗流方程组解的整体存在与爆破

本文研究了由边界条件耦合的多维牛顿渗流方程组解的长时间行为.利用构造的多种上下解,得到了整体存在临界曲线与Fujita临界曲线.     

非线性Schrdinger方程的整体吸引子(英文)

本文讨论了一类带调和势｜x｜2的非线性Schrdinger方程解的长时间行为,证明了整体吸引子的存在性.     

三维Keller-Segel-Navier-Stokes方程弱解的整体存在性

本文证明具有logistic源的一个3维Keller-Segel-Navier-Stokes方程弱解的整体存在性,并研究了弱解的长时间行为.     

边界耦合的牛顿渗流方程组解的整体存在与爆破（英文）

本文研究了由边界条件耦合的多维牛顿渗流方程组解的长时间行为.利用构造的多种上下解,得到了整体存在临界曲线与Fujita临界曲线.     

Kolmogorov-Spieqel-Siveshinky方程大时间问题的Fourier拟谱逼近

该文讨论了Kolmogorov-Spieqel-Siveshinky方程的周期初值问题, 研究了半离散Fourier拟谱解的长时间行为, 证明了半离散系统的收敛性和整体吸引子的存在性. 构造了全离散的三层显式Fourier拟谱格式, 并证明了该格式的收敛性, 最后通过数值计算验证了格式的可信性. 数值结果表明: 该格式是长时间稳定并可取时间大步长. 作者模拟了方程的解在相空间的轨线, 得到了一些有意义的结论.     

非线性Schr(o)dinger方程的整体吸引子

本文讨论了一类带调和势|x|2的非线性Schr(o)dinger方程解的长时间行为,证明了整体吸引子的存在性.     

由各向异性逆高斯曲率流支配的凸超曲面的发展运动

本文研究了一个n维欧氏空间中的凸初始超曲面的各向异性曲率流,其包含了高斯曲率,支撑函数和其梯度的函数.在适当的假设下,我们给出了该流的长时间存在性和收敛性的证明.作为推论,我们给出对偶Orlicz-Minkowski问题解的存在性.     

三维Keller-Segel-Navier-Stokes方程弱解的整体存在性

本文证明具有logistic源的一个3维Keller-Segel-Navier-Stokes方程弱解的整体存在性，并研究了弱解的长时间行为.     

An approximation of stochastic hyperbolic equations: case with Wiener process

In the present paper, the two‐step difference scheme for the Cauchy problem for the stochastic hyperbolic equation is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of four problems for hyperbolic equations are obtained. The theoretical statements for the solution of this difference scheme are supported by the results of the numerical experiment. Copyright © 2012 John Wiley & Sons, Ltd.     

Long Time Asymptotic Behavior of Solution of Difference Scheme for a Semilinear Parabolic Equation

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as $t\rightarrow\infty$. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.     

一类非线性离散系统的指数二分性及其在数值分析和计算中的应用

本文中我们考虑一类二阶非线性常微分方程的边值问题的迎风差分格式.我们运用奇异摄动方法构造了该迎风差分方程解的渐近近似,并利用指数二分性理论证明了有一个低阶方程其解是该迎风方程式的在边界外的一个良好近似.我们还构造了校正项,使校正项与低阶方程的解之和是一个渐近近似.最后一些数值例子用于显示本文方法的应用.     

On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation

We study a nonlocal boundary value problem for a degenerating pseudoparabolic third-order equation of the general form. For the solution of the problem, we obtain a priori estimates in differential and difference form, which imply the stability of the solution with respect to the initial data and right-hand side on a layer as well as the convergence of the solution of the difference problem to the solution of the differential problem.     

Numerical solution by the quasi-reversibility method of unstable problems for the first-order evolution equation

Prior bounds are derived on the solution of the perturbed problem in different versions of the quasi-reversibility method used for approximate solution of unstable problems for first-order evolution equations. An example of such a problem is provided by the problem backward in time for the equation of heat conduction. Approximate solution of perturbed problems by difference methods is considered. The investigation of the difference schemes of the quasi-reversibility method relies on the general theory of p-stability of difference schemes. Specific features of solution of problems with non-self-adjoint operators are considered. Efficient difference schemes are constructed for multidimensional problems.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 93–124, 1993.     

Finite difference method of first boundary problem for quasilinear parabolic systems (V)

Iterative difference schemes for the first boundary problem of the quasilinear parabolic system are established and the convergence of the difference solution for the iterative difference schemes to the unique solution of the problem is proved. Project supported by the National Natural Science Foundation of China and the Foundation of CAEP.     

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.     

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

In this paper, we study the approximation of reverse parabolic problem with integral boundary condition. The Rothe difference scheme for an approximate solution of reverse problem is discussed. We establish stability and coercive stability estimates for the solution of the Rothe difference scheme. In sequel, we investigate the first order of accuracy difference scheme for approximation of boundary value problem for multidimensional reverse parabolic equation and obtain stability estimates for its solution. Finally, we give numerical results together with an explanation on the realization in one- and two-dimensional test examples.     

三阶线性变系数差分方程的Mikusiski算符解法（Ⅲ）

本文在［３］，［４］工作的基础上，利用变数算符的思想以及Ｍｉｋｕｓｉｎｓｋｉ算符域中移动算符和变系数移动算符级数的有关结果，解决了一般的三阶线性变系数差分方程的求解问题，并且绘出了一些特殊的三阶线性变系数差分方程的更好的解式；此外，还试图为实现更高阶线性变系数差分方程的求解提供思想方法。     

Poincaré方程的线性差耦合系统

该文对Poincare方程的线性差耦合系统的同相解及反相解进行研究,得到了同相解稳定的参数区域,并在对角线性差耦合的情形下.对其反相解的存在性及稳定性进行了完整的分析,改进了文[1]的结果.