A singular perturbation of the heat equation with memory

In this paper we consider a hyperbolic equation, with a memory term in time, which can be seen as a singular perturbation of the heat equation with memory. The qualitative properties of the solutions of the initial boundary value problems associated with both equations are studied. We propose numerical methods for the hyperbolic and parabolic models and their stability properties are analyzed. Finally, we include numerical experiments illustrating the performance of those methods.     

The Uniform <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Behavior for Time Discretization of an Evolution Equation

The uniform L^2 stability and convergence properties for the time discretization of an evolution equation with a memory term are studied.The methods are based on the second-order backward difference methods.The memory term is approximated by the second-order convolution quadrature and interpolant quadrature.     

Invariance principles for hybrid systems with memory

Hybrid systems with memory are dynamical systems exhibiting both delayed and hybrid dynamics. Such systems can be described by hybrid functional inclusions. Classical invariance principles play an instrumental role in proving stability and convergence of dynamical systems. Invariance principles for general hybrid systems with delays, however, remain an open topic. In this paper, we prove invariance principles for hybrid systems with memory, using both Lyapunov–Razumikhin function and Lyapunov–Krasovskii functional methods. These invariance principles are then applied to derive two stability results as corollaries.     

On Multistep Methods for Differential Equations of Fractional Order

This paper concerns with numerical methods for the treatment of differential equations of fractional order. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which order conditions and stability properties are investigated. Dedicated to the memory of Professor Aldo Cossu     

THE EFFECT OF MEMORY TERMS IN DIFFUSION PHENOMENA

In this paper the effect of integral memory terms in the behavior of diffusion phenomenais studied.The energy functional associated with different models is analyzed and stabilityinequalities are established.Approximation methods for the computation of the solutionof the integro-differentiaI equations are constructed.Numerical results are included.     

The Asymptotic Behavior for Numerical Solution of a Volterra Equation

Long-time asymptotic stability and convergence properties for the numerical solution of a Volterra equation of parabolic type are studied.The methods are based on the first-second order backward difference methods.The memory term is approximated by the comvolution quadrature and the interpolant quadrature.Discretization of the spatial partial differential operators by the finite element method is also considered.     

Stability analysis for networked control systems with sampling,transmission protocols and input delays

In this paper, the stability problem is investigated for networked control systems. Input delays and multiple communication imperfections containing time-varying transmission intervals and transmission protocols are considered. A unified framework based on the hybrid systems with memory is proposed to model the whole networked control system. Hybrid systems with memory are used to model hybrid systems affected by delays and permit multiple jumps at a jumping instant. The stability analysis depends on the Lyapunov–Krasovskii functional approaches for hybrid systems with memory and the proposed stability theorem does not need strict decrease of the Lyapunov–Krasovskii functional during jumps. Based on the developed stability theorems, stability conditions for networked control systems are established. An explicit formula is given to compute the maximal allowable transmission interval. In the special case that the networked control system contains linear dynamics, an explicit Lyapunov functional is constructed and stability conditions in terms of linear matrix inequalities (LMI) are proposed. Finally, an example of a chemical batch reactor is given to illustrate the effectiveness of the proposed results.     

A class of nonlinear differential equations with fractional integrable impulses

In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki’s normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki–Ulam’s type stability are introduced and generalized Ulam–Hyers–Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.     

Stability analysis of block boundary value methods for neutral pantograph equation

This paper deals with the convergence and stability properties of block boundary value methods (BBVMs) for the neutral pantograph equation. Due to its unbounded time lags and limited computer memory, a change in the independent variable is used to transform a pantograph equation into a non-autonomous differential equation with a constant delay but variable coefficients. It is shown under the classical Lipschitz condition that a BBVM is convergent of order p if the underlying boundary value method is consistent with order p. Furthermore, it is proved under a certain condition that BBVMs can preserve the asymptotic stability of exact solutions for the neutral pantograph equation. Meanwhile, some numerical experiments are given to confirm the main conclusions.     

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Derivative free iterative methods with memory of arbitrary high convergence order

In this article we present three derivative free iterative methods with memory to solve nonlinear equations. With the process developed, we can obtain n-step derivative free iterative methods with memory of arbitrary high order. Numerical examples are provided to show that the new methods have an equal or superior performance, on smooth and nonsmooth equations, compared to classical iterative methods as Steffensen’s and Newton’s methods and other derivative free methods with and without memory with high order of convergence.     

Exponential stabilization of a flexible structure with a local interior control and under the presence of a boundary infinite memory

In this paper, we deal with the interior stabilization problem of a flexible structure governed by a hyperbolic partial differential equation coupled to two ordinary differential equations. Contrary to the previous works on the system, the boundary control is subject to the presence of an infinite memory term. In order to deal with such a nonlocal term, the minimal state approach is invoked. Specifically, a localized interior control is proposed in order to compensate the infinite memory effect. Thereafter, reasonable assumptions on the memory kernel are evoked so that the closed-loop system is shown to be well-posed thanks to semigroups theory of linear operators. Furthermore, the resolvent method is used to establish the exponential stability of the system.     

具有时滞的双向联想记忆(BAM)的神经网络的全局动力学行为

在没有假定关联函数的光滑性,单调性和有界性的条件下,应用Liapunov泛函方法和矩阵代数技术,得到具有常数传输时滞的双向联想记忆(BAM)的神经网络模型平衡点存在性和全局指数稳定性的一些新的充分条件,这些条件可以由网络参数,连接矩阵和关联函数的Lipschitz常数所表示的M矩阵来刻化．这些结果不仅是简单和实用的,而且相对于已有文献的结果具有较少的限制和更易于验证．     

基于LMSV模型的半参数方法对股市波动长记忆特征的识别

在金融时间序列波动具有显著的长记忆性这一背景之下,研究了LMSV模型长记忆参数的估计问题。首先,分析了LMSV模型的相关性质;接着,根据LMSV模型和ARFIMA模型的良好对应关系,提出了估计LMSV模型长记忆参数的半参数方法;最后,基于股市数据,验证了波动半参数方法的有效性。     

Leakage delays in BAM

A well known bidirectional associative memory (BAM) model is generalized with the introduction of discrete time delays in the leakage (or forgetting) terms. By using a model transformation, the system is converted to one of a neutral delay system. Two sets of delay dependent sufficient conditions are derived for the existence of a unique equilibrium as well as its asymptotic and exponential stability. The methods of degenerate Lyapunov-Kravsovskii functionals and inequalities together with some properties of M-matrices are used in the derivation of sufficient conditions. In the absence of leakage delays, the sufficient conditions lead to some known sufficient conditions.     

The most storage economical Runge-Kutta methods for the solution of large systems of coupled first-order differential equations

It is shown how the attainable minimum for the memory requirements of Runge-Kutta methods can be realised for methods of the third order. These economisable third order methods belong to a one parameter sub-family from which two particular members with low error bound are selected.     

Two-parameter families of predictor-corrector methods for the solution of ordinary differential equations

Two-parameter families of predictor-corrector methods based upon a combination of Adams- and Nyström formulae have been developed. The combinations use correctors of order one higher than that of the predictors. The methods are chosen to give optimal stability properties with respect to a requirement on the form and size of the regions of absolute stability. The optimal methods are listed and their regions of absolute stability are presented. The efficiency of the methods is compared to that of the corresponding Adams methods through numerical results from a variable order, variable stepsize program package.     

求解约束优化问题的记忆梯度Goldstein-Lavintin-Polyak投影算法

给求解无约束规划问题的记忆梯度算法中的参数一个特殊取法,得到目标函数的记忆梯度G o ldste in-L av in tin-Po lyak投影下降方向,从而对凸约束的非线性规划问题构造了一个记忆梯度G o ldste in-L av in tin-Po lyak投影算法,并在一维精确步长搜索和去掉迭代点列有界的条件下,分析了算法的全局收敛性,得到了一些较为深刻的收敛性结果.同时给出了结合FR,PR,HS共轭梯度算法的记忆梯度G o ldste in-L av in tin-Po lyak投影算法,从而将经典共轭梯度算法推广用于求解凸约束的非线性规划问题.数值例子表明新算法比梯度投影算法有效.     

An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations

This paper deals with stability of the extended Runge–Kutta methods for nonlinear neutral delay-integro-differential equations. The stability results in the reference [Y. Yu, L. Wen, S. Li, Nonlinear stability of Runge–Kutta methods for neutral delay integro-differential equations, Appl. Math. Comput. 191 (2007) 543–549] are improved. With this improvement, several new numerical stability criteria are obtained, it is proven that the extended Runge–Kutta methods are globally and asymptotically stable under the suitable conditions.     

广义神经网络及其稳定性,容错性分析

本文建立了一个广义神经网络模型，并研究了它的渐近稳定性和指数稳定性，由这些结果我们可以估计各记忆模式的吸引域及其中每一点趋向记忆模式的指数收敛速度，以此来评价网络的容错能力．