Asymptotic solutions and error estimates for linear systems of difference and differential equations

Classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant, i.e., k+o(1) as t tends to infinity. Here we are interested in more precise information about the o(1) terms, specifically how they depend precisely on certain perturbation terms in the equation. Results along these lines were given by Gel'fond and Kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations.     

Almost diagonal systems of linear difference equations

Linear homogeneous difference systems of equations on an infinite interval are considered. Conditions are given which allow to represent a fundamental solution as a product of two square matrices. The product is made of a diagonal matrix and a matrix which is a perturbation of the identity. The diagonal matrix being on the left in the proposed product rather than on the right as it has been traditionally represented. A self contained theorem of asymptotic approximation is provided which is applied to a special family of difference systems. A theorem, utilizing a linear transformation, which is best possible in a certain sense, is also given.     

On the reducibility of countable systems of linear difference equations

We solve the problem of reducibility of a countable linear system of standard difference equations with unbounded right-hand sides by the method of construction of iterations with accelerated convergence. For systems of this type with bounded right-hand sides, this problem is reduced to a finite-dimensional case.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1533–1541, November, 1995.     

Perturbations for linear difference equations

In this paper we study boundary value problems for perturbed second-order linear difference equations with a small parameter. The reduced problem obtained when the parameter is equal to zero is a first-order linear difference equation. The solution is represented as a convergent series in the small parameter, whose coefficients are given by means of solutions of the reduced problem.     

Characterization of regular singular linear systems of difference equations

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=. We are concerned with the behavior of solutions near the pointz= (the only fixed singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system () we can assign a number , called the Moser's invariant of (), so that the system is regular singular if and only if 1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an irreducible form. The computation ot the number can be done explicitly from this irreducible form.     

Oscillation for linear non-autonomous systems of difference equations with continuous arguments

In this paper, we establish sufficient conditions for the oscillation of the linear non-autonomous systems of difference equations with continuous arguments

     

Variation of parameters formulae with initial time difference for linear integrodifferential equations

This article investigates the relationship between an unperturbed integrodifferential system and a perturbed integrodifferential system that have different initial positions and an initial time difference (ITD). Variation of parameter techniques are employed to obtain integral formulae and to define Lyapunov-like functions.     

Asymptotic analysis for linear difference equations

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

     

A posteriori error estimates for linear equations

Summary This paper describes upper and lowerp-norm error bounds for approximate solutions of the linear system of equationsAx=b. These bounds imply that the error is proportional to the quantity wherer is the residual andq is the conjugate index top. The constant of proportionality is larger than 1 and lies in a specified range. Similar results are obtained for approximations toA ^–1 and solutions of nonsingular linear equations on general spaces.Research was partially supported by NSF Grant DMS8901477     

Exponential dichotomy of systems of linear difference equations with periodic coefficients

We study the problem of exponential dichotomy for the systems of linear difference equations with periodic coefficients. Some criterion is established for exponential dichotomy in terms of solvability of a special boundary value problem for a system of discrete Lyapunov equations. We also give estimates for dichotomy parameters.     

Reducibility of linear systems of difference equations with almost periodic coefficients

We establish sufficient conditions of the reducibility of the linear system of difference equationsx(t+1)=Ax(t) + P(t) x(t) with an almost periodic matrixP(t) to a system with a constant matrix.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1661–1667, December, 1993.     

Minimal universal denominators for linear difference equations

We provide minimal universal denominators for linear difference equations with fixed leading and trailing coefficients. In the case of first-order equations, they are factors of Abramov's universal denominators. While in the case of higher order equations, we show that Abramov's universal denominators are minimal.     

Initial-value problems for linear partial difference equations

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions _i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.     

On difference Riccati equations and second order linear difference equations

In this paper, we treat difference Riccati equations and second order linear difference equations in the complex plane. We give surveys of basic properties of these equations which are analogues in the differential case. We are concerned with the growth and value distributions of transcendental meromorphic solutions of these equations. Some examples are given.     

Second order linear difference equations

We provide the explicit solution of a general second order linear difference equation via the computation of its associated Green function. This Green function is completely characterized and we obtain a closed expression for it using functions of two–variables, that we have called Chebyshev functions due to its intimate relation with the usual one–variable Chebyshev polynomials. In fact, we show that Chebyshev functions become Chebyshev polynomials if constant coefficients are considered.