基于补偿倒向Euler方法带分数Brown运动的随机固定资产模型数值解的均方散逸性

讨论了一类带分数Brown运动随机固定资产模型数值解的均方散逸性.在漂移系数和扩散系数满足单边Lipschitz条件和有界条件下,建立了随机固定资产模型补偿倒向Euler法数值解均方散逸性的判定准则.最后通过数值算例对结论进行了验证.     

随机种群模型数值解的均方散逸性

讨论了随机种群模型数值解的均方散逸性,基于步长受限制和无限制的两种条件,利用补偿的和无补偿的数值方法研究了随机种群模型数值解的均方散逸性.从而得出补偿的数值算法更适合解决随机种群模型数值解的均方散逸性问题.     

非线性多比例延迟微分方程向后Euler方法的散逸性

本文讨论了多比例延迟微分方程的散逸性，证明了应用向后Euler方法求解多比例延迟微分方程数值解仍保持散逸性，它可视为文献[9]中相应结果的推广。     

带跳和分数Brown运动的固定资产系统倒向Euler数值解的均方渐近有界性

介绍了一类与年龄相关的随机固定资产系统倒向Euler数值解法.漂移系数和扩散系数在单边Lipschitz条件和有界条件下,建立了随机固定资产系统倒向Euler数值解均方渐近有界性的判定准则.最后通过数值算例对结论进行了验证.     

随机资产系统补偿倒向Euler数值解的渐近有界性

首先给出了一类带分数Brown运动的固定资产系统,并给出了相应的补偿倒向Euler法.其次,在漂移系数满足单边Lipschitz条件,且扩散系数满足有界条件下,建立了补偿倒向Euler数值解均方渐近有界性的判定准则.最后通过算例对文章的结论进行了验证.     

线性中立型随机延迟微分方程Euler方法的均方稳定性

本文讨论Euler方法用于求解线性中立型随机延迟微分方程初值问题时数值解的稳定性,利用了一种不同于以往文献中的证明技巧,给出了Euler方法均方稳定的一个充分条件.文末的数值试验证实了本文所获理论结果的正确性.     

具有Markov调制的随机年龄结构种群系统半驯服Euler法的指数稳定性

根据半驯服Euler法讨论了具有Markov调制的随机年龄结构种群系统的数值解.在非局部Lipschitz条件下,利用Burkholder-Davis-Gundy不等式、Ito公式和Gronwall引理,证明了半驯服Euler数值解不仅强收敛阶数为0.5,而且这种方法在时间步长一定的条件下有很好的均方指数稳定性.最后通过数值例子对所给的结论进行了验证.     

具有Markov调制的随机种群系统半驯服Euler法的指数稳定性

根据半驯服Euler法讨论了具有Markov调制的随机年龄结构种群系统的数值解. 在非局部Lipschitz条件下, 利用~Burkholder-Davis-Gundy~不等式、It\^{o} 公式和~Gronwall~引理, 证明了半驯服Euler数值解不仅强收敛阶数为~0.5, 而且这种方法在时间步长一定的条件下有很好的均方指数稳定性. 最后通过数值例子对所给的结论进行了验证.     

Cahn-Hilliard方程的高阶保能量散逸性方法

能量散逸性是物理和力学中某些微分方程一项重要的物理特性.构造精确地保持微分方程能量散逸性的数值格式对模拟具有能量散逸性的微分方程具有重要的意义.本文利用四阶平均向量场方法和傅里叶谱方法构造了Cahn-Hilliard方程高阶保能量散逸性格式.数值结果表明高阶保能量散逸性格式能很好地模拟Cahn-Hilliard方程在不同初始条件下解的行为,并且很好地保持了Cahn-Hilliard方程的能量散逸特性.     

分段连续型随机微分方程指数Euler方法的稳定性

给出了线性分段连续型随机微分方程指数Euler方法的均方指数稳定性.经典的对稳定性理论分析,通常应用的是Lyapunov泛函理论,然而,应用该方程本身的特点和矩阵范数的定义给出了该方程精确解的均方稳定性.以往对于该方程应用隐式Euler方法得到对于任意步长数值解的均方稳定性,而应用显式Euler方法得到了相同的结果.最后,给出实例验证结论的有效性.     

Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching

In this paper, we concentrate on the numerical approximation of solutions of stochastic delay integro-differential equations with Markovian switching (SDIDEsMS). We establish the split-step backward Euler (SSBE) scheme for solving linear SDIDEsMS and discuss its convergence and stability. Moreover, the SSBE method is convergent with strong order γ = 1/2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable and general mean-square stable are obtained. Some illustrative numerical examples are presented to demonstrate the stability of the numerical method and show that SSBE method is superior to Euler method.     

Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations

This paper deals with the mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. It is shown that the stochastic theta methods inherit the mean-square exponential stability property of the underlying system. Moreover, the backward Euler method is mean-square exponentially stable with less restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps

Numerical Algorithms - This article aims to reveal the mean-square convergence rate of the backward Euler method (BEM) for a generalized Ait-Sahalia interest rate model with Poisson jumps. The main...     

Convergence to equilibrium for a time semi-discrete damped wave equation

We prove that the solution of the backward Euler scheme applied to a damped wave equation with analytic nonlinearity converges to a stationary solution as time goes to infinity. The proof is based on the {\L}ojasiewciz-Simon inequality. It is much simpler than in the continuous case, thanks to the dissipativity of the scheme. The framework includes the modified Allen-Cahn equation and the sine-Gordon equation.     

Adaptive control of linear systems with coordinatewise-parametric perturbations of white noise type

Conditions for the mean-square dissipativity of adaptive stabilization systems for a linear object under coordinate-parametric perturbations of white noise type are obtained. A linear adaptive regulator with adjustable coefficients is chosen. For adjusting parameters, an adaptation algorithm is synthesized by the passification method. The number of inputs in objects under consideration may differ from that of outputs. The proof is based on the construction of a quadratic stochastic Lyapunov function. (In the case of purely parametric perturbations, the obtained conditions are known to be necessary and sufficient for the existence of a Lyapunov function with these properties.) Dissipativity conditions for the constructed closed system are obtained; it is shown that, in some special cases, the dissipativity of the closed system is preserved under white-noise perturbations of any intensity.     

Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay

In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L² dissipative for any time step size and H¹ dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017     

带分数布朗Brown的随机比例方程半隐式Euler法的数值解(英文)

本文给出并分析了Poisson随机跳测度驱动的带分数Brown运动的随机比例方程半隐式Euler法的数值解,在局部Lipschitz条件下,证明了在均方意义下半隐式Euler数值解收敛到精确解.     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.