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.     

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.     

On numerical solutions for hyperbolic–parabolic equations with the multipoint nonlocal boundary condition

A numerical method is proposed for solving multi-dimensional hyperbolic–parabolic differential equations with the nonlocal boundary condition in t and Dirichlet and Neumann conditions in space variables. The first and second order of accuracy difference schemes are presented. The stability estimates for the solution and its first and second orders difference derivatives are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of a one-dimensional hyperbolic–parabolic differential equations with variable coefficients in x and two-dimensional hyperbolic–parabolic equation.     

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.     

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.     

Optimal boundary control of heat conduction problems on an infinite time domain by control parameterization

The present paper deals with an optimal boundary control problem in which the process of systems under consideration is governed by a linear parabolic partial differential equation over an infinite time interval. The objective of the paper is to determine the optimal boundary control that minimize a given energy-based performance measure. The performance measure is specified as a quadratic functional of displacement and a suitable penalty term involving the boundary controls. In order to determine the optimal boundary controls, the problem with boundary controls are converted into a problem with distributed controls. The modal space technique is then used to reduce the system into the optimal control of time invariant lumped parameter system. The associated system of uncoupled first order initial value problems is solved in terms of controllers. Next step deals with the computation of the control and trajectory of the linear time-invariant lumped parameter. For this we approximate the controllers by a finite number of orthogonal exponential zero-interpolants over the interval [0,∞). The resultant performance index after using the optimality condition leads to a system of linear algebraic equations. The suggested technique is easy to implement on digital computer. We provide a numerical example to demonstrate the applicability and efficiency of the proposed approach.     

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。     

Representation and realization of operational differential equations with time-varying coefficients

An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.     

Laxe-Milgram定理的一个非线性变体在椭圆型p-Laplace问题上的应用

许多物理学和工程学的问题中,它们的模型都是一些偏微分方程加上适当的边值条件和初始值条件,并且以等式形式出现。Lax-Milgram定理就是解决此类问题的一种方法,Lax--Milgram定理要求事先给定的空间是HiIbert空间,但在实际应用中并非都是Hilbert空间,而有一类问题要求事先给定的空间是自反的Banach空间,本文研究了Lax-Milgram定理的一个非线性变体在椭圆型p-Laplace问题上的应用。     

An analytical solution for the multidimensional degradation of a packed bed thermocline

A multidimensional, time-dependent model of the thermocline degradation in a packed bed thermal storage tank is presented. The formulation includes the effects of finite tank length, multidimensional thermal conduction, heat transfer between the fluid and solid portions of the bed, and heat losses across all tank surfaces. The technique used to solve the coupled pair of parabolic partial differential equations describing the degradation of the thermocline is based on a vectorized separation of variables approach. Assuming cylindrical geometry, the analysis leads to separate eigenvalue problems for the radial, angular, and axial spatial dependencies of the fluid and solid temperatures in the bed. The eigenvalues are readily calculated and the corresponding eigenfunctions are shown to form a complete set of spatial basis functions for the solution space. The method and results are illustrated by several examples.     

Consensus control for multi-agent systems with distributed parameter models via iterative learning algorithm

This paper deals with the problem of iterative learning control algorithm for a class of multi-agent systems with distributed parameter models. And the considered distributed parameter models are governed by the parabolic or hyperbolic partial differential equations. Based on the framework of network topologies, a consensus-based iterative learning control protocol is proposed by using the nearest neighbor knowledge. When the iterative learning control law is applied to the systems, the consensus errors between any two agents on L² space are bounded, and furthermore, the consensus errors on L² space can converge to zero as the iteration index tends to infinity in the absence of initial errors. Simulation examples illustrate the effectiveness of the proposed method.     

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。     

Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems

The design problem of collocated feedback controllers is addressed in this paper for a class of semi-linear distributed parameter systems described by parabolic partial differential equation (PDE), where a finite number of local actuators and sensors are intermittently distributed in space. A Lyapunov direct method for the exponential stability analysis of the resulting closed-loop system is first presented for the system, in which the first mean value theorem for integration and the Wirtinger's inequality are employed. The corresponding stabilization condition is then derived through the analysis result. Finally, the proposed design method is implemented on the feedback control of a fisher equation and its effectiveness is evaluated through simulation results.     

On the Computation of Functionals Over Sample Paths of a Randomly Rotating Rigid Body

This paper deals with the computation of the values of two functionals which are defined over the sample paths of a randomly rotating rigid body. It is assumed that the body is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the body's motion, two functionals are defined over its sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations is suggested, and its efficiency and applicability are demonstrated with examples.     

Nonlinear dynamic modeling and responses of a cable dragged flexible spacecraft

The space debris removal system (SDRS) of tethered space tug is modelled as a cable dragged flexible spacecraft. The main goal of this paper is to develop a dynamic modeling approach for mode characteristics analysis and forced vibration analysis of the planar motion of a cable dragged flexible spacecraft. Solar arrays of the spacecraft are modelled as multi-beams connected by joints with additional rotating spring where the nonlinear stiffness, damping and friction are considered. Using the Global mode method (GMM), a novel analytical and low-dimensional nonlinear dynamic model is developed for vibration analysis of SDRS to enhance the design capacity for better fulfillment of space tasks. The linear and nonlinear partial differential equations that governing transverse vibration of solar arrays, transverse and longitudinal vibrations of cable are derived, along with the matching and boundary conditions. The natural frequencies and analytical global mode shapes of SDRS are determined, and orthogonality relations of the global mode shapes are established. Dynamical equations of the system are truncated to a set of ordinary differential equations with multiple-DOF. The validity of the method is verified by comparing the natural frequencies obtained from the characteristic equation with those obtained from FEM. Interesting mode localization and mode shift phenomena are observed in mode analysis. Dynamic responses of the system excitated by fluctuation of attitude control torque and short-time attitude control torque are worked out, respectively. Nonlinear behaviors are observed such as hardening, jump and super-harmonic resonances. Residual vibration of the overall system with considering the varous values of nonlinear stiffness, damping coefficient and friction coefficient has shown that the nonlinearity of joints has a great influence on the vibration of the overall system.     

An optimum technique for determining elastic continuum dynamic response using modal analysis

A modal analysis method which minimizes a weighted error function for the differential equation of an elastic continuum is presented. The technique essentially determines optimum time functions for the modal amplifiers by a simple extremization step. The method reduces to a differential equation collocation method when the error weighting function is a Dirac delta function over the continuum space. By means of an example, the technique is shown to have the potential advantages of modal decoupling despite modal non-orthogality, increased accuracy and simplified implementation compared to a more standard approach such as Lagrange's equations. The method is shown to be equivalent to Lagrange's equations for the special case of orthogonal modes.     

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.     

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。     

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.     

纵向槽孔板的应力集中系数

根据Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ原理，建立了一种具有一个无外力圆形边界的特殊杂交应力元，其应力场准确满足弹性力学平衡方程、协调方程及无外力圆弧上的边界条件。根据修正的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ原理，建立了另一种具有一个无外力直边界的特殊杂交应力元，其应力场在元上变分满足平衡方程，但准确满足无外力直边上的边界条件。数值算例表明：特殊假定应力元与一般假定位移元这两类元联合，能十分有效地进行纵向倒圆槽孔板的应力集中分析。