Solutions Of Linear Differential-difference Equations Research Articles

MODELING STABILITY OF DIFFERENTIAL-DIFFERENCE EQUATIONS WITH DELAY

Differential-difference and differential-functional equations are mathematical models of ma\-ny applied problems in automatic control and management systems, chemical, biological, technical, economic and other processes whose evolution depends on prehistory. In the study of the problems of stability, oscillation, bifurcation, control, and stabilization of solutions of linear differential-difference equations, the location of the roots of the corresponding characteristic equations is very important. Note that there are currently no effective algorithms for finding the zeros of quasipolynomials. When studying the approximation of a system of linear differential-difference equations, it was found that the approximation of nonsymptotic roots of their quasi-polynomials can be found with the help of characteristic polynomials of the corresponding approximating systems of ordinary differential equations . This paper investigates the application of approximation schemes for differential-difference equations to construct algorithms for the approximate finding of nonsymptotic roots of quasipolynomials and their application to study the stability of solutions of systems of linear differential equations with many delays. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer. Computational experiments on special test examples showed the high efficiency of the proposed algorithms for studying the stability of linear differential-difference equations.

Growth of solutions of linear differential-difference equations with coefficients having the same logarithmic order

In this paper, we investigate the relations between the growth of meromorphic coefficients and that of meromorphic solutions of complex linear differential-difference equations with meromorphic coefficients of finite logarithmic order. Our results can be viewed as the generalization for both the cases of complex linear differential equations and complex linear difference equations.

Necessary and Sufficient Conditions for the Absolute Instability of Solutions of Linear Differential-Difference Equations with Self-Adjoint Operator Coefficients

For linear differential-difference equations of retarded and neutral types with infinitely many deviations and self-adjoint operator coefficients, we present necessary and sufficient conditions for the absolute instability of the trivial solutions.

Approximation of periodic solutions of linear differential-difference equations of neutral type by spline-collocation method

We construct a method for the approximation of periodic solutions of linear differential-difference equations of the neutral type by using cubic splines and investigate conditions for the convergence of the proposed approximation scheme.

Topics in pharmacokinetics, II: Identification of time-lag processes

We discuss some aspects of the asymptotic behavior of solutions of linear differential-difference equations arising in two-compartment models where time lags exist, and present a method for the determination of rate constants and delays on the basis of observations over time.

On periodic solutions of linear differential-difference equations with constant coefficients

On periodic solutions of linear differential-difference equations with constant coefficients

Properties of solutions of linear differential-difference equations

Properties of solutions of linear differential-difference equations

On the solution of linear differential-difference equations

On the solution of linear differential-difference equations

A Time Optimal Control Problem for Systems Described by Differential Difference Equations

The aim of this paper is to establish the solution of an optimal time control problem for a physical system whose state is described by a linear differential-difference equation with retarded argument. We have obtained here a generalisation of the results of Bellman and his collaborators Glicksberg, Gross and Kalaba, and of LaSalie and Neustadt by using a technique due to LaSalle, with the help of the kernel matrix representation of Bellman and Cooke and also a new integral representation for the solutions of linear differential-difference equations.

Asymptotic behavior of solutions of differential-difference equations.

Abstract : In this paper, the problem was considered of determining the asymptotic behavior of solutions of linear differentialdifference equations whose coefficients possess asymptotic series. Although the problem is considerably more complicated than the corresponding problem for ordinary differential equations, by means of a sequence of transformations the problem was reduced to a form where the standard techniques of ordinary differential equation theory could be employed. The differential-difference equation was transformed into an integral equation which was trans formed into an integro- differential equation. At this point the Liouville transformation plays a vital role. Although the guiding ideas were simple, the analysis became formidable. For this reason, only some of the more immediate aspects of the problem were considered.

Asymptotic series for the solutions of linear differential-difference equations

Abstract : The problem of determining the asymptotic nature of the solution of linear differential-difference equations of the form x'(t) = A(t)x(t) + B(t)x(t-1), where the coefficients A(t) and B(t) possess asymptotic series expansions has previously been treated only under quite special conditions and by quite complicated methods. The purpose of this paper is to present a new technique which will yield more comprehensive results. To illustrate the method, unhindered by analytic details, the author considers the scalar version of the foregoing equation and treat only the asymptotic series of the solution associated with the characteristic root of the largest real part.