Linear Differential-difference Equations Research Articles

Analytical kinetics of clustering processes with cooperative action of aggregation and fragmentation.

Some models of clustering processes are formulated and analytically solved employing generating functions methods. Those models include events that result from combined action of the coagulation and fragmentation processes. Fragmentation processes of two kinds, so-called similar and arbitrary, ones, are brought forward, and the explicit forms of their solutions are produced. This implies some possibility of existence of different aggregation mechanisms for clusters creation differing in their inner structure. All the models are based on "the three-level bunch" scheme of interaction between the system states. Those states are described in terms of the probability to find the system in the state with an exactly given number of clusters. The models are linear in the probability functions due to the assumption that the rates of elementary acts are permanent. Some peculiarities of application of the generating function method to solution of the linear differential-difference equations are revealed. The illustration of the problem in terms of a traffic jam picture is not a specific one.

CHATTER PREDICTION OF END MILLING IN A VERTICAL MACHINING CENTER

The development of a high-speed and high-precision vertical machining center had been executed as a national project in Korea. In the machining center, the phenomenon of chatter vibration was an important factor, and should be examined for developing it by a relevant way. In a high-speed and high-precision vertical machining center, in order to observe and predict the chatter vibration, firstly, modal testing was performed to obtain modal parameters of the system. Secondly, in order to predict the cutting forces for end-milling process under various cutting conditions, a simplified model was presented. Finally, the chatter prediction for the vertical machining center was formulated as linear differential–difference equations, and verified by the cutting tests. The method in this work will provide an engineer who designs a machining center with a tool to predict the chatter phenomenon.

Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials

We define a special type of reduction in a free left module over a ring of difference–differential operators and use the idea of the Gröbner basis method to develop a technique that allows us to determine the Hilbert function of a finitely generated difference–differential module equipped with the natural double filtration. The results obtained are applied to the study of difference–differential field extensions and systems of difference–differential equations. We prove a theorem on difference–differential dimension polynomial that generalizes both the classical Kolchin’s theorem on dimension polynomial of a differential field extension and the corresponding author’s result for difference fields. We also determine invariants of a difference–differential dimension polynomial and consider a method of computation of the dimension polynomial associated with a system of linear difference–differential equations.

Implicit linear time-dependent differential-difference equations and applications

The paper deals with the differential-difference equation Σ [A j (t)u'(t - ω j ) + B j (t - ω j )] = f(t) i=0 in a Banach space. The operator coefficient A 0 (t) of the delay-free derivative is allowed to be degenerate. Existence and uniqueness theorems are proved under the main assumption that for every t ≥ 0 the point μ = 0 is a polar singularity of the resolvent (A 0 (t) + μB 0 (t)) -1 . The results are applied to evolution problems of microwave circuits.

Similar Operators and a Functional Calculus for the First-Order Linear Differential Operator

The general problem of finding solutions of linear functional equations is undoubtedly one of the main problems of applied mathematics. Among such equations the classes of linear differential equations and linear difference equations are very important and numerous theories and meth- ods for their solution have been developed. The methods based on integral transforms and the operational methods are certainly the ones used most often. In the present paper we present some general linear algebra results that may be considered as a unified approach to the transform and operational methods. We use the linear algebra foundations of the Laplace transform and the concept of similarity to generalize a description of the general solution of linear differential equations with constant coefficients in terms of certain basis for the space of exponential polynomials, and a convolution product defined in an algebraic way using the basis. The generalization gives an explicit construction of the general solution of any * Research partially supported by a fellowship from CONACYT-Mexico. ´ † Research partially supported by a grant from CONACYT-Mexico. E-mail: verde@xan- ´

The Fredholm Alternative for Functional Differential Equations of Mixed Type

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.

Complex magnetic susceptibility of uniaxial superparamagnetic particles in a strong static magnetic field

The magnetic relaxation of a system of single-domain ferromagnetic particles in the presence of a strong static magnetic field directed at an arbitrary angle relative to the particle anisotropy axis is investigated. A system of linear difference-differential equations for the moments (averaged spherical harmonics) is derived without recourse to the Fokker-Planck equation by averaging Gilbert’s equations with a fluctuating field. An exact solution (in terms of matrix continuous fractions) is found for this system. The relaxation times and spectra of the complex magnetic susceptibility are calculated.

De l'impact du progrès technique sur la croissance dans un modèle à générations de capital

This paper focuses on the impact of technical progress on growth in the framework of a vintage capital model. This structure of capital allows for creative destruction, in the sense that the obsolescence of capital results from embodied technical progress in new equipments. The rate of growth is endogenous in Harrod-Domar's way and depends on a rate of technical progress which remains exogenous, contrary to what happens in endogenous growth theory. The model reduces to a system of linear differential-difference equations. Under a certain level of technical progress, the rate of growth is asymptotically constant. Above it, the model oscillates around a stationary state. Even if its impact is positive in a partial equilibrium context, an increase in the rate of technical progress has a negative impact on growth in a general equilibrium framework.

A Difference Differential Equation of Euler–Cauchy Type

We study a class of advanced argument linear difference differential equations analogous to Euler–Cauchy ordinary differential equations. Solutions of two equations of this type have arisen as adjoint functions in sieve theory, and they are also of use in control theory. Here we study the problem in a general setting. Subject to mild assumptions, each of our equations is shown to have a unique solution which is analytic in the right half-plane. In some cases the solution is a polynomial, and in others it has an asymptotic expansion. Finally, the solution is shown to have a representation as an exponential of a Hellinger type integro-differential operator acting on a monomial.

Robust Stability of Differential-Difference Systems with Linear Time-Delay

Abstract A system of linear differential-difference equations with constant coefficients and one linearly increasing delay is considered. Some algebraic type necessary and sufficient conditions of the asymptotic stability and instability of the system are obtained. A criterion of the robust stability is also given. Special classes of uncertain systems with linear delays, for which the testing set consists of a finite number of elements are derived.

Global stability of sets for linear

Stability of sets of general type is investigated with respect to linear impulsive differential-difference equations with variable impulsive perturbations. The main results are proved with the aid of piecewise continuous functions, which are analogues of the Lyapunov functions.

Basic Theory of Singular Systems of Linear Differential Difference Equations

This paper discusses singular systems of constant coefficient linear differential difference equations with delay, there are many cases of this class of systems because of different coefficients, in some cases, there exists a unique solution, and in some cases, there exist infinite solutions, under some initial condition and consistency condition; In some cases, there is no solution. Also there are many types of equation included in this class of systems, such as, neutral differential difference equation, composite systems of differential algebraic equation with delay, and so on. Above all, it is very complicated. This paper analyzes the existence, uniqueness of solutions, establishes the basic theory of solutions.

Global attractivity of linear non-autonomous neutral differential-difference equations

In this paper, by constructing Liapunov functionals, we study the global attractivity of linear non-autonomous neutral differential difference equation $$\tfrac{d}{{dt}}\left[ {x(t) - a(t)x(t - \tau )} \right] = - \mu (t)x(t) - b(t)x(t - \sigma ) + c(t)x(t - \gamma ),$$ (1) and obtain some new easily-checked sufficient conditions for the global attractivity of the zero solution of equation (1).

Reaction-diffusion processes of hard-core particles

We study a 12-parameter stochastic process involving particles with two-site interaction and hard-core repulsion on ad-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes, and other processes, the stochastic variables are particle occupation numbers taking valuesn x=0,1. We show that on a ten-parameter submanifold thek-point equal-time correlation functions 〈n x1...n xk〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n x〉 satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum HamiltonianH, the model becomes equivalent to a lattice model in thermal equilibrium ind+1 dimensions. We show that the spectrum ofH is identical to the spectrum of the quantum Hamiltonian of ad-dimensional anisotropic, spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system.

Experimental Verification of a Stability Theory for Periodic Cutting Operations

The stability of periodic cutting operations, the dynamics of which are described by linear differential-difference equations with periodic coefficients, is studied. A new stability theory that uses parametric transfer functions and Fourier analysis to obtain the characteristic equation of such systems is experimentally verified. The theory is applied to single-point turning of a compliant work piece with two degrees-of-freedom. The theoretical predictions of both the critical depth of cut for chatter-free turning and the corresponding chatter frequency were found to be in good agreement with the measurements obtained from actual chatter tests under various surface speeds.

Stability for Differential Difference Equations

In this paper, we consider linear autonomous differential difference equations of retarded or neutral type. Some algebraic sufficient conditions for the zero solution to be stable (hyperbolic) for all the values of the delays are given. And some simple criteria for the asymptotically stable cone to be convex are obtained.

Tracking in matrix systems

This paper provides conditions, and a sense, in which the matrix of a system of linear differential or linear difference equations can be assumed to be similar to a diagonal matrix.

Periodic solutions of non linear differential difference equations

In this paper we will be interested in the behaviour of long chains of coupled gravitational pendula. We will prove existence and uniqueness of periodic solutions for such chains under periodic forcing and will prove that under some smoothness assumptions the chain behaves as an uncoupled one. We will also analyse a more general class of differential difference equations and prove existence and unicity results for periodic solutions.

Stability with probability 1 of solutions of systems of linear stochastic differential-difference Ito equations

Conditions for absolute (independent of lags) asymptotic stability with probability 1 of systems of stochastic equations cited in the title of this paper are obtained. The proposed approach allows us to reduce the problem of analyzing stability to determination of the conditions for the existence of a positively determined solution of a linear matrix equation. These conditions are stated in terms of locations of eigenvalues of the matrix constructed from the elements of the matrices of the initial system.

Quadratic Optimization for Infinite-Dimensional Linear Differential Difference Type Systems: Syntheses via the Fredholm Equation

A closed-loop solution for quadratic optimization is presented for control systems whose model is given by infinite-dimensional linear differential difference equations: \[\frac{{dx(t)}}{{dt}} = Ax(t) + \sum_{i = 1}^n {A_i x\left( {t - h_i } \right)} + Bu(t) + \sum_{j = 1}^m {B_j u\left( {t - r_j } \right)} .\] A nonsemigroup evolution formula for the fundamental solution is given first. The system is then reduced to a Volterra integral system. By the semicausality approach, a new optimality principle is established, and the closed-loop optimal control is given by real-time state feedback plus retarded state integral feedback. The feedback operator is determined by solving a linear Fredholm integral equation. The two crucial methods, semicausal dynamical optimization for Volterra integral systems and nonsemigroup evolution property, are brought together to generalize the syntheses results to infinite-dimensional cases.