Formulation of stable difference schemes for systems of initial-value partial differential equations

A general system of initial-value partial differential equations is classified into four categories based on the partial differential operators which define the equations. Specific combinations of the operators are termed “invariants” since they are common to all finite difference approximations to the system of equations. The “invariants” are used to a priori determine if one may formulate a stable difference approximation to a system of partial differential equations. This is in essence a numerical existence theory.     

对称扰动理论在偏微分方程中的应用研究

非线性科学已经被广泛应用于数学、物理、化学、经济等领域。许多非线性现象都可以用非线性偏微分方程来很好地描述,所以得到非线性偏微分方程的解具有重要的意义。在研究非线性科学的同时,出现了一些带有扰动项的非线性偏微分方程。为了研究这种扰动偏微分方程,一些以对称理论为基础的扰动方法相继产生。本文主要研究对称扰动理论在偏微分方程中的应用,寻求偏微分方程的近似对称约化和无穷级数解。     

Optimal control of stochastic functional neutral differential equations with time lag in control

In this work, we consider an optimal control problem of a class of stochastic differential equations driven by additive noise with aftereffect appearing in control. We develop a semigroup theory of the driving deterministic neutral system and identify explicitly the adjoint operator of the corresponding infinitesimal generator. We formulate the time delay equation under consideration into an infinite dimensional stochastic control system without time lag by means of the adjoint theory established. Consequently, we can deal with the associated optimal control problem through the study of a Hamilton–Jacob–Bellman (HJB) equation. Last, we present an example whose optimal control can be explicitly determined to illustrate our theory.     

Numerical solution of systems of mixed partial differential equations associated with transport phenomena with exchange

The so-called “percolation operations” which involve simultaneous heat and mass transport are defined in terms of a system of mixed partial differential equations which are nonlinear and include variable coefficients. Second-order interpolation followed by integration over an increment of the time or space variable is proposed as a method for arriving at finite difference equation equivalents of the original partial differential equations. A model change is also used to give physical significance to finite difference equivalents associated with derivatives in the space dimension. It is shown that second-order interpolation has the advantage over linear interpolation that it introduces a factor β which in a stability analysis based on cencentrations offers a range of choices that will insure stable calculation on a digital machine.     

高斯平滑算子对Y—K模型改进的图像去噪方法

研究了在偏微分方程理论框架下进行图像去噪的方法,重点对二阶和四阶偏微分方程的主要去噪方法进行了分析。二阶偏微方程模型中的P—M模型能很好地去除噪声,但时常会出现块状现象;四阶偏微分方程模型中Y—K模型能消除阶梯效应但会出现斑点现象。使用Gilboa扩散系数、中值滤波器和高斯平滑算子对YK模型进行改进。实验证明．新模型减少了迭代次数,提高了算法效率。也一定程度上避免了块状及斑点现象。     

Effect of Non-Darcian Flow on Time Rate of Consolidation

The effect of non-Darcian flow on the consolidation behavior of clay soils is studied, and its role in the extrapolation of laboratory test results to field problems is evaluated. This is accomplished by postulating a reasonably general four-parameter velocity-gradient relationship which, by proper choice of parameters, is capable of characterizing much of the published experimental data; then, this relationship is combined with the standard assumptions of classical consolidation theory to develop a nonlinear parabolic partial differential equation, which is solved by use of a finite difference technique. The stability and convergence criteria for related linear and quasi-linear equations are empirically extended to the associated nonlinear equations, and a comparison is made between various explicit and implicit finite difference schemes, with the result that a sufficiently accurate and more economical numerical solution is obtained by use of an explicit scheme. Typical solutions for various specific cases confirm and offer an explanation for the well-known phenomenon wherein the time rate of consolidation is found to decrease as the load increment decreases; also, the thickness of the consolidating layer is shown to affect the dimensionless time rate of consolidation. These conditions indicate that laboratory consolidation test results can be applied to a field situation only if appropriate stress and thickness corrections are made.     

常微分方程教学体系改革初探

在常微分方程课程教学内容和体系的研究与实践中,对常微分方程的理论、实践、教学方法作了深入讨论,进而构思出常微分方程课程体系框架。     

On optimal location and management of a new industrial plant: Numerical simulation and control

Within the framework of numerical modelling and multi-objective control of partial differential equations, in this work we deal with the problem of determining the optimal location of a new industrial plant. We take into account both economic and ecological objectives, and we look not only for the optimal location of the plant but also for the optimal management of its emissions rate. In order to do this, we introduce a mathematical model (a system of nonlinear parabolic partial differential equations) for the numerical simulation of air pollution. Based on this model, we formulate the problem in the field of multi-objective optimal control from a cooperative viewpoint, recalling the standard concept of Pareto-optimal solution, and pointing out the usefulness of Pareto-optimal frontier in the decision making process. Finally, a numerical algorithm – based on a characteristics/Galerkin discretization of the adjoint model – is proposed, and some numerical results for a hypothetical situation in the region of Galicia (NW Spain) are presented.     

时动边界上热通量重构正问题求解方法

具有Neumann边界条件的抛物型方程的初边值问题是偏微分方程研究领域的一类经典的问题。正问题是由已知的边界条件和初始条件来求区域温度场的问题。若所给边界是固定区域的称为定边界问题,而现实中又有一类问题其边界随时间变化,这样的问题称为动边界问题。本文对于时动边界上的热通量重构正问题的求解提出了人工边界的方法。在人工边界的基础上采用了位势理论方法求解此定解问题,并与前面已经给出的差分方法进行了比较。为了检验方法的可行性和两种方法的优劣,给出了数值模拟。     

Stabilization of time delay systems with saturations via PDE predictor boundary control design

This paper addresses the stabilization issue of linear time delay system with input saturation and distinct input delays via predictor feedback boundary control algorithm by employing transport partial differential equations (PDEs). First, the addressed ordinary differential equation (ODE) system with input delay is equivalently represented as a cascade of an ODE and transport PDEs. Second, by employing the backstepping Volterra integral transformation technique, the equivalent cascade system is transformed into a stable target system, whose kernels are solved by the constraints satisfying transport PDEs. Third, based on the boundary conditions of the obtained invertible transformation, the proposed feedback control law can be formulated. Fourth, by applying semigroup operator theory, the well-posedness of the resulting system is proved and consequently, novel exponential stability conditions of the addressed system are established. Then, the domain of attraction region under the given actuator saturation constraints is estimated with the help of the solution of obtained stability conditions. Finally, a demonstrative simulation example is offered to verify the feasibility and usefulness of the results.     

Switching fault-tolerant control of a moving vehicle-mounted flexible manipulator system with state constraints

In this brief, a switching fault-tolerant control (FTC) scheme is presented for a moving vehicle-mounted flexible manipulator subject to state constraints. The dynamic characteristics of the system are represented by coupled ordinary differential equations and partial differential equations (ODEs–PDEs). When actuators are healthy, vibration control and position regulations can be realized without violation of the given constraints based on a Barrier Lyapunov Function (BLF). Moreover, a switching strategy is introduced to prevent the transgression of constraints even under actuator failure by detecting actuator faults as-assisted by the proposed monitoring functions. The closed-loop states are kept within the given bounds under FTC laws. By extending LaSalle's Invariance Principle to an infinite dimension, the asymptotic stability of the fault-free closed-loop system is strictly verified. Simulation results demonstrate the effectiveness of the proposed approach.     

An adaptive mesh method with variable relaxation time

Moving mesh partial differential equations have been widely used in the last decade for solving differential equations exhibiting large solution variations such as shock waves and boundary layers.In this paper, we have applied a dynamic adaptive method for solving time-dependent differential equations. The mesh velocities are governed by an equation in which a relaxation time is employed to move nodes in such a way that they remain concentrated in regions of rapid variation of the solution. A numerical example involving a blow-up problem shows the advantage of using a variable relaxation time over a fixed one.     

非线性发展方程非守恒格式的计算稳定性

针对非线性发展方程的非守恒格式 ,以二维非线性浅水波方程为例 ,给出了计算稳定的必要性条件 .在数值试验的基础上 ,进一步讨论了非线性发展方程非守恒格式与初值之间的关系 .理论分析和数值试验证明 ,非守恒格式的计算稳定性不仅与格式的结构有关 ,而且还由初值及其偏导数的形式所决定     

On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only

This study deals with the stability analysis of a flexible structure with one and only one boundary control. The system is composed of three parts: a cart (motorized platform), a flexible cable, and a load mass attached to the lower part of the cable. This situation leads to a hybrid system as a mathematical model for the cable dynamics: one partial differential equation coupled to two ordinary differential equations. Despite the presence of a time-delay in the top-end of the cable, we are able to prove that the hybrid system is well-posed in the sense of semigroups theory and more importantly, only one boundary control can guarantee the exponentially decay of the energy of the system under reasonable conditions on the parameters of the system. This outcome considerably improves the result recently established in [17], where two more controls are required: one interior (Kelvin–Voigt) damping which acts over the entire cable and another boundary control which is exerted on the lower-end of the cable. Furthermore, we provide an estimate of the exponential decay of the system by means an appropriate Lyapunov functional. Lastly, numerical examples are presented in order to ascertain and highlight our theoretical outcomes.     

Uniform separable differential operators with parameters

In this paper we study boundary value problems for anisotropic partial differential-operator equations with parameters. The principal part of the appropriate differential operators are not self-adjoint. Several conditions for the uniform separability in weighted Banach-valued L_p-spaces are given. Sharp estimates for the resolvent of the corresponding differential operator are obtained. In particular the positivity and R-positivity of these operators are established. As an application we study the separability of degenerate DOEs, maximal regularity for degenerate abstract parabolic problem with parameters, the uniform separability of finite and infinite systems for degenerate anisotropic partial differential equations with parameters.     

Stagnation point flow toward a stretching/shrinking sheet with a convective surface boundary condition

The steady two-dimensional stagnation point flow toward a stretching/shrinking sheet with the bottom surface of the sheet heated by convection from a hot fluid is considered. The governing partial differential equations are transformed into ordinary differential equations, before being solved numerically. Results for the skin friction coefficient and the local Nusselt number as well as the temperature profiles are presented for different values of the governing parameters. Effects of the governing parameters on the heat transfer characteristics are thoroughly examined. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique.     

A short account of the theory of lubrication. III. dimensional theory,with applications

In Part I properties of the scale coördinate, of the form: B(n + θ) are discussed. n is shown to be associated with the operation of counting scale marks, θ with the operation of estimating between them, and B, with the operational and configurational aspects of that part of apparatus which lies adjunct to the scale system.In Part II three types of measurement codification are discussed: (a) the differential interval; (b) the finite amorphous interval; (c) the scale interval; a relationship among them is postulated.In Part III the finite differences in scale coördinates are defined and simple theorems are used to illustrate these definitions. Simple difference equations in scale coördinates are solved to illustrate macroscopic “selection principles” arising partly out of the methodology of codifying a coincidence in scale coördinates.In Part IV an example of causally related dimensional systems is described by use of the scale coördinate. This example is taken from the perfect gas law and Van der Waals' gas law.     

Stability in mean of partial variables for stochastic reaction–diffusion systems with Markovian switching

This paper is denoted to investigating stability in mean of partial variables for stochastic reaction–diffusion equations with Markovian switching (SRDEMS). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations with Markovian switching (SODEMS) and using Itô formula, sufficient criteria on uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, exponential stability in mean of partial variables for SRDEMS are first derived. An example is presented to illustrate the effectiveness and efficiency of the obtained results.     

Approximate solution (with error bounds) to a nonlinear,nonautonomous second-order differential equation

Two approximations are developed to the solution of an important nonlinear, nonautonomous second-order differential equation that arises in various fields of science and technology such as operations research, mathematical ecology and epidemiology. The origin of the second-order differential equation from a system of two nonlinear first-order differential equations modelling, for example, Lanchester-type combat between two homogeneous military forces is discussed. Extension of our results to a more general system of nonlinear first-order differential equations is indicated. Error bounds that do not require that the exact solution be known are developed. Some connections between our results and those for the Liouville-Green (or WKB) approximation to the solution of the linear second-order equation are indicated.     

启示性方法用于分析有限差分方程的计算稳定性（英文）

这篇文章通过一些典型例子讨论了在用启示性方法时从原偏微分方程推导来的稳定性条件与从差分方程展开式推导来的稳定性条件间的不同点。结果表明，对于部分有限差分方程，在用启示性方法分析其计算稳定性的过程中最好采用从差分方程推导来的展开式以期得到较合理的结果。在文章的另一部分，反证法的运用表明了从启示性方法推导来的稳定性条件并非全都是必要条件，在应用中应引起注意。