求解时滞微分方程组的Rosenbrock方法的GP-稳定性

讨论了求解延时微分方程组的Rosenbrock方法的数值稳定性,分析了求解线性试验方程组的Rosenbrock方法的稳定性态,并证明了数值求解延时微分方程组的Rosenbrock方法是GP-稳定的充分必要条件是Rosenbrock方法是A-稳定的．     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.     

Numerical solution of two-sided space-fractional wave equation using finite difference method

In this paper, a class of finite difference method for solving two-sided space-fractional wave equation is considered. The stability and consistency of the method are discussed by means of Gerschgorin theorem and using the stability matrix analysis. Numerical solutions of some wave fractional partial differential equation models are presented. The results obtained are compared to exact solutions.     

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

This paper is motivated by the stability problem of nonconstant periodic solutions of time‐periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions that are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way.     

A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters

In this work, we propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, which goes beyond the existing theory, and for the special case of period-two parameters we analytically show the conjecture is true. For this case we show that the stability region in parameter space obtained from the conjecture is larger than a previously proposed stability region. The period-three case is investigated numerically and similar extensions are realized. This suggests that the current theory cited in this paper, while giving sufficient conditions for stability is far from optimal.     

具有限时滞一阶线性泛函微分方程的稳定性区域划分

讨论了一阶线性有限时滞泛函微分方程的稳定性区域,用一个超平面把参数空间划分为不同的稳定性区域.这个超平面上的每一点对应于特征方程在纯虚轴上至少存在一个零根(原点除外),所得结论可用于Hofp分枝分析和控制理论.     

Convergence Rate to Stationary Solutions for Boltzmann Equation with External Force

For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. (2005) through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

A reduced-order extrapolating space–time continuous finite element method for the 2D Sobolev equation

This paper mainly concerns with the order reduction to the coefficient vectors of the classical space–time continuous finite element (STCFE) solutions for a two-dimensional Sobolev equation. The classical STCFE model is first constructed for the governing equation, and the theoretical results of the existence, stability, and convergence are provided for the STCFE solutions. We then employ a proper orthogonal decomposition to develop a reduced-order extrapolating STCFE (ROESTCFE) vector model with the lower dimension, and demonstrate the existence, stability, and convergence for the ROESTCFE solutions by the matrix means, resulting in the very concise and flexible theoretical analysis. Lastly, we examine the effectiveness of the developed ROESTCFE model by several numerical tests. It is shown that the ROESTCFE method is computationally very cheap in actual applications.     

一类单参数发展方程的时间周期解的分歧及其稳定性

本文应用渐近开和谱分析的方法，对发展方程的时间周期解的分歧及其稳定性进行了讨论，得到了存在性和稳定性判据的结果．     

无阻尼振动方程解的稳定性

在无干扰力的环境中,定性分析无阻尼振动方程解的稳定性,可归结为下述几个问题:牛顿第二运动定律应用于振动建模描述力与运动的关系,线性化方程是求解微分方程的有效方法;单摆振动的等时性与非等时性特征表明,微分方程的解不仅决定于方程本身,而且也决定于解的初值;微分方程定性理论,特别是李雅普诺夫第二方法,是研究非线性微分方程解的稳定性的有效手段;如何构造李雅普诺夫函数,至今仍是一个吸引人的研究课题.     

Stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions

In this paper, we study the stability of nonlinear impulsive stochastic differential equations in terms of two measures. The concept of perturbing Lyapunov functions is introduced to discuss stability properties of solutions of nonlinear impulsive stochastic differential equations in terms of two measures. By using perturbing Lyapunov functions and comparison method, some sufficient conditions for the above stability are given.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

随机时滞2D-Navier-Stokes方程的指数稳定性

由于流体受到某些遗传和不确定信息外力的影响,考虑了含时变时滞随机外力的2D-Navier-Stokes方程.借助随机分析中的It6公式和Burkholder-Davis-Gundy不等式,证明了大粘性系数情形方程整体弱解的均方指数稳定和几乎必然指数稳定.     

Convex solutions of polynomial-like iterative equations

In this paper convex solutions and concave solutions of polynomial-like iterative equations are investigated. A result for non-monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones. Furthermore, another condition for convex solutions, which is weaker in some aspects, is also given. The uniqueness and stability of those solutions are also discussed.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.     

非Lipschitz条件下由Lévy过程驱动的倒向随机微分方程解的存在唯一及其稳定性

本文研究了由满足某种矩条件下Lévy过程相应的Teugel鞅及与之独立的布朗运动驱动的倒向随机微分方程,给出了飘逸系数满足非Lipschitz条件下解的存在唯一及稳定性结论.解的存在性是通过Picard迭代法给出的.解的L2收敛性是在飘逸系数弱于L2收敛意义下所得到的.     

A stability criterion for a neutral difference equation with delay

A stability criterion for a neutral difference equation with delay is established which extends and improves a result of Ladas et al. [1]. Our derivation is based on Lyapunov's direct method for stability, and avoids the approach employed by Ladas et al. who had considered asymptotic behaviors of oscillatory and nonoscillatory solutions.     

AN ITERATIVE EQUATION ON THE UNIT CIRCLE

A functional equation of nonlinear iterates is discussed on the circle S1 for its continuous solutions and differentiable solutions. By lifting to R, the existence, uniqueness and stability of those solutions are obtained. Techniques of continuation are used to guarantee the preservation of continuity and differentiability in lifting.