An Operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow

In this paper we present a simple, general methodology for the generation of high-order operator decomposition (splitting) techniques for the solution of time-dependent problems arising in ordinary and partial differential equations. The new approach exploits operator integration factors to reduce multiple-operator equations to an associated series of single-operator initial-value subproblems. Two illustrations of the procedure are presented: the first, a second-order method in time applied to velocity-pressure decoupling in the incompressible Stokes problem; the second, a third-order method in time applied to convection-Stokes decoupling in the incompressible Navier-Stokes equations. Critical open questions are briefly described.     

Discretized LKF method for stability of coupled differential‐difference equations with multiple discrete and distributed delays

Time‐delay systems described by coupled differential‐functional equations include as special cases many types of time‐delay systems and coupled differential‐difference systems with time delays. This article discusses the discretized Lyapunov–Krasovskii functional (LKF) method for the stability problem of coupled differential‐difference equations with multiple discrete and distributed delays. Through independently dividing every delay region that the plane regions consists in two delays to discretize LKF, the exponential stability conditions for coupled systems with multiple discrete and distributed delays are established based on a linear matrix inequality (LMI). The numerical examples show that the analysis limit of delay bound in which the systems are stable may be approached by our result. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of Impulsive Stochastic Neutral Partial Differential Equations With Infinite Delays

In this paper, we study the existence and asymptotic stability in the pth moment of the mild solutions to impulsive stochastic neutral partial differential equations with infinite delays. Sufficient conditions ensuring the stability of the impulsive stochastic system are established. The results are obtained via the Banach fixed point theorem.     

水平阻尼作为干扰的平台式惯导统一控制方程

研究惯导系统统一控制的优化设计。传统的统一控制采用一体化设计方案,通用性不强。为简捷高效地实现惯导计算机的程序编排,提出了将水平阻尼作为干扰项的平台式惯导统一控制。通过改变系统框图结构,将无阻尼惯导系统和阻尼网络人为分离,水平阻尼网络的输出可视为干扰项。系统控制方程始终采用无阻尼条件下的表达式,惯导系统控制和阻尼网络可以实现模块化设计,使控制的实现形式和推导过程大为简化。给出了惯导计算机的运算流程,并分别在静、动基座条件下,对无阻尼、内水平阻尼和外水平阻尼状态的惯导系统做了仿真。结果表明新的统一控制模型有效体现了惯导系统的参数变化,且设计简便,具有更好的实用性。     

A continuous-time framework for least squares parameter estimation

This paper proposes a continuous-time framework for the least-squares parameter estimation method through evolution equations. Nonlinear systems in the standard state space representation that are linear in the unknown, constant parameters are investigated. Two estimators are studied. The first one consists of a linear evolution equation while the second one consists of an impulsive linear evolution equation. The paper discusses some theoretical aspects related to the proposed estimators: uniqueness of a solution and an attractive equilibrium point which solves for the unknown parameters. A deterministic framework for the estimation under noisy measurements is proposed using a Sobolev space with negative index to model the noise. The noise can be of large magnitude. Concrete signals issued from an electronic device are used to discuss numerical aspects.     

FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations

In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n$n$ independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented.     

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier–Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier–Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity–stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.     

Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators

We consider infinite-dimensional port-Hamiltonian systems described on jet bundles. Based on a power balance relation we introduce the port-Hamiltonian system representation using differential operators regarding the structural mapping, the dissipation mapping and the input mapping. In contrast to the well-known representation on the basis of the underlying Stokes–Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations. The presented constructions will be specialized to mechanical systems to which class also the presented examples belong.     

Modelling the dynamics of the students’ academic performance in the German region of the North Rhine-Westphalia: an epidemiological approach with uncertainty

Student academic underachievement is a concern of paramount importance in Europe, where around 15% of the students in the last high school courses do not achieve the minimum knowledge academic requirement. In this paper, we propose a model based on a system of differential equations to study the dynamics of the students’ academic performance in the German region of the North Rhine-Westphalia. This approach is supported by the idea that both good and bad study habits, are a mixture of personal decisions and influence of classmates. This model allows us to forecast the student academic performance by means of confidence intervals over the next few years.     

PHASE-SPACE DIFFUSION EQUATIONS FOR SINGLE BROWNIAN PARTICLE MOTION NEAR SURFACES

In numerous physical processes involving the motion of micron and submicron sized particles near surfaces, such as the filtration of hydrosols and aerosols, the particle motion is the net result of the combined effects of fluid convection, external forces, particle inertia, Brownian particle motion, and particle-surface fluid dynamic interactions. The most general method of describing particle motion under the combined action of these effects is through the so-called Fokker-Planck equation. In the absence of particle-surface fluid dynamic interactions, the Fokker-Planck equation is well-known, and it has been applied in a general way to problems involving the adsorption or deposition of Brownian particles onto surfaces through a solution technique known as the Brownian dynamics simulation method.

In this study, the Fokker-Planck equation for Brownian particle motion near surfaces is generalized to include particle-surface fluid dynamic interactions. The Fokker-Planck equation is shown to follow from the Liouville equation for the Brownian particle and n-fluid molecules present in the system, thus, establishing a firm theoretical foundation for the Fokker-Planck equation and the various other phase-space diffusion equations that follow from it.

Based on diagonalization of the Fokker-Planck equation, its short-time behavior is also derived here which enables a generalization of the Brownian dynamics method for the study of particle motion near surfaces including fluid dynamic interactions. Additionally, a perturbation solution of the Fokker-Planck equation under the conditions of small, but finite particle Stokes number is also derived. These solutions are shown to agree with previously given representations of the Smoluchowski or convective-diffusion equation for Brownian particle motion near surfaces, as well as with inertial corrections to the Smoluchowski equation available in the literature. This latter equation is also generalized here to include particle-surface fluid dynamic interactions.