Geometric Integration Methods that Preserve Lyapunov Functions

We consider ordinary differential equations (ODEs) with a known Lyapunov function V. To ensure that a numerical integrator reflects the correct dynamical behaviour of the system, the numerical integrator should have V as a discrete Lyapunov function. Only second-order geometric integrators of this type are known for arbitrary Lyapunov functions. In this paper we describe projection-based methods of arbitrary order that preserve any given Lyapunov function. AMS subject classification (2000) 65L05, 65L06, 65L20, 65P40     

自全国各地100多位理事Road Machinery Branch of China Highway and Transportation Society He

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.     

几类特定边界值条件下的常微分方程特征值问题

在文献[1]的基础上,借助于图形分析方法,讨论了几类特定边界值条件下的特征值问题.同时,本文对文献[1]中的错误结论作出了更正.本文给出的方法对讨论非线性边值问题的可解性提供了一种新的途径.     

氯化钾水溶液微观结构的分子动力学模拟研究

氯化钾水溶液不同于氯化钠的,其在低温下只能形成无水盐和冰.现有研究表明在氯化钠水溶液中不仅存在着水分子连续分布的区域与离子团簇,而且这二者的物质组成也与该溶液所形成晶体的分别对应.为了探寻其他溶液中是否存在此类对应关系,并考察溶液的微观结构,本文采用分子动力学方法对氯化钾水溶液进行了研究,表明了该溶液的一些性质.氯化钾水溶液中K⁺-K⁺和Cl^--Cl^-径向分布函数的特征具有一致性,峰的最大值所对应的位置都分别相同,明显不同于氯化钠水溶液的.系列时刻下瞬态图像内O到其最近离子距离中最大值的统计结果表明氯化钾水溶液中存在着一定大小的水分子连续分布的区域,其平均尺寸至少为2.26 nm;瞬态图像中K⁺与其最近邻Cl^-之间的距离主要分布在0.28 nm～0.38 nm之间,占比约为97.4%;溶液中存在着较大和较小两类离子团簇,较大团簇的平均尺寸为1.73nm,平均离子数是25.0,其内部的离子与周围离子之间具有与氯化钾晶体类似的结构;这些结果表明氯化钾水溶液中也...     

Periodic solutions of a periodic stochastic human immunodeficiency virus model with distributed delay and cytotoxic T lymphocytes immune response

In this paper, a periodic stochastic human immunodeficiency virus (HIV) model with distributed delay and cytotoxic T lymphocytes (CTL) immune response is investigated. First, by It $ô$'s formula, we show that the solution with any positive initial value is global and positive. Then, by the stochastic comparison theorem, we obtain the sufficient conditions guaranteeing the existence and global attractivity of infection-free periodic solution. Furthermore, we discuss the existence of the infective periodic solution by Has'minskii theory. Finally, numerical examples are given to illustrate the results.     

Explicit solutions of Boussinesq--Burgers equation

Darboux transformation with multi-parameters for the Boussinesq--Burgers (B--B) equation is derived. For an application, some important explicit solutions of the B--B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.     

Numerical solutions to the time-dependent Bloch equations revisited

The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch equations. First, the time-dependent Bloch equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. The validity of this method was investigated by comparing with the analytical solutions in the case of constant radiofrequency irradiation. There was a good agreement between them, indicating the validity of this method. As a further example, this method was applied to the time-dependent Bloch equations in the two-pool exchange model for chemical exchange saturation transfer (CEST) or amide proton transfer (APT) magnetic resonance imaging (MRI), and the Z-spectra and asymmetry spectra were calculated from their solutions. They were also calculated using the fourth/fifth-order Runge-Kutta-Fehlberg (RKF) method for comparison. There was also a good agreement between them, and this method was much faster than the RKF method. In conclusion, this method will be useful for analyzing the complex CEST or APT contrast mechanism and/or investigating the optimal conditions for CEST or APT MRI.     

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.     

自相似分形上的非线性抛物方程(英文)

本文研究了分形区域上的一类半线性抛物方程.对于分形上恰当定义的拉普拉斯算子,建立了抛物方程的比较原理.利用上下解方法,我们得到了在一定条件下半线性抛物方程正解的存在性和唯一性.