Numerical Approximation of Neutral Differential Equations on Infinite Interval

In this paper we study numerical approximation of linear neutral differential equations on infinite interval using equations with piecewise constant arguments. As an application of our approximation results, we obtain stability theorems for some classes of linear delay and neutral difference equations.     

Remarks on the Notion of Order of Difference Equations

In the paper, the notion of order of a difference equation is introduced in such a way that this order is invariant with respect to the change of the independent variable. For the general case, a formula for the general solution of linear difference equation of k -th order is given. It is shown that, in contrast to differential equations, the dimension of the linear space of solutions of linear homogenous difference equation can be lowered if their domain of definition is restricted appropriately.     

Difference Equations and Lettenmeyer's Theorem

Many classical results for ordinary differential equations have counterparts in the theory of difference equations, although, in general, the technical details for the difference versions are more involved than the corresponding ones for differential equations. This note surveys material related to a difference analogue of Lettenmeyer's theorem. The projection method of Harris et al. , developed to treat certain questions in the analytic theory of ordinary differential equations is used to obtain counterparts for linear difference equations as well as extensions to certain nonlinear differential and difference equations.     

On Invariance Factors and Invariance Vectors for Difference Equations

In this paper, the concept of invariance factors and invariance vectors to obtain invariants (or first integrals) for difference equations will be presented. It will be shown that all invariance factors and invariance vectors have to satisfy a functional equation. This concept turns out to be analogous to the concept of integrating factors and integrating vectors for ordinary differential equations.     

Exact Solutions to a Finite-difference Model of a Nonlinear Reaction-advection Equation: Implications for Numerical Analysis

This article gives exact solutions to a finite-difference model of a nonlinear reaction-advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.     

Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

Indefinite integrals of special functions from hybrid equations

ABSTRACT

Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.     

Nonlinear Stochastic Difference Equations Driven by Martingales

Abstract

We provide in this paper a systematic development of nonlinear stochastic difference equations driven by martingales (that depend on a spatial parameter); three such equations are considered. We begin with the existence and uniqueness of solutions and continue with the study of stochastic properties, such as the martingale and Markov properties, along with ? irreducibility and recurrence. We discuss in the final section the discrete-time flow and asymptotic flow properties of the solution process.     

Classification of Nonoscillatory Solutions of a Higher Order Neutral Difference Equation

In this paper, classification schemes are given for the nonoscillatory solutions of higher order neutral difference equations. Several necessary and/or sufficient conditions are obtained for the existence of each kind of solution.     

Global Behaviour of Solutions of Nonlinear Delay Difference Equations*

In this paper global asymptotic stability and asymptotic behaviour of solutions of nonlinear delay difference equations has been studied and a few sets of sufficient conditions for global asymptotic stability are derived.     

On Detecting Solutions of Polynomial Nonlinear Difference Equations

In this paper a method for discovering solutions of nonlinear polynomial difference equations is presented. It is based on the concepts of i -operator and star-product. These notions create a proper algebraic background by means of which we can find linear equations "included" into the original nonlinear one and to seek for solutions among them. A corresponding algorithm and some examples are also provided.     

A DISCRETE-TIME ITÔ'S FORMULA

Abstract

The numerical methods on stochastic differential equations (SDEs) have been well established. There are several papers that study the numerical stability of SDEs with respect to sample paths or moments. In this paper, we study the stability in distribution of numerical solution of SDEs.     

On the Periodic Structure of Delayed Difference Equations of the Form x n = f ( x n − k ) on I and S 1

The aim of this paper is to give an account of some results recently obtained in Combinatorial Dynamics and apply them to get for k S 2 the periodic structure of delayed difference equations of the form x n = f ( x n m k ) on I and S 1 .     

Numerical Solution to Hybrid Stochastic Differential Systems

Abstract

In this article numerical methods for solving hybrid stochastic differential systems of Itô-type are developed by piecewise application of numerical methods for SDEs. We prove a convergence result if the corresponding method for SDEs is numerically stable with uniform convergence in the mean square sense. The Euler and Runge–Kutta methods for hybrid stochastic differential equations are specifically described and the order of the error is given for the Euler method. A numerical example is given to illustrate the theory.     

Models and numerical schemes for generalized van der Pol equations

This paper presents three generalizations of the van der Pol equation (VDPE) using newly proposed three new generalized K-, A- and B-operators. These operators allow kernel to be arbitrary. As a result, these operators provide a greater generalization of the VDPE than the fractional integral and differential operators do. Like the original VDPE, the generalized van der Pol equations (GVDPEs) are also nonlinear equations, and in most cases, they can not be solved analytically. Numerical algorithms are presented and used to solve the GVDPEs. Results for several examples are presented to demonstrate the effectiveness of the numerical algorithms, and to examine the behavior of the GVDPEs and the limit cycles associated with them. Although the numerical algorithms have been used to solve the GVDPEs only, they can also be used to solve many other generalized oscillators and generalized differential equations. This will be considered in the future.     

Difference Equations and Divisibility Properties of Sequences

There are many different ways of defining a sequence in terms of solutions to difference equations. In fact, if a sequence satisfies one recurrence then it satisfies an infinite number of recurrences. Arithmetic properties of an integral sequence are often studied by direct methods based on the combinatorial or algebraic definition of the numbers or using their generating function. The rational generating function is the main tool in obtaining various difference equations with coefficients and initial values exhibiting divisibility patterns that can imply particular arithmetic properties of the solutions. In this process, we face the challenging task of finding difference equations that are relevant to the divisibility properties by transforming the original rational generating function. As a matter of fact, it is not necessarily the simple difference equation that helps the most in proving the properties. We illustrate this process on several examples and a sequence involving a p -sected binomial sum of the form y n = y n ( p , a )= ~ k =0 X n kp a k where p is an arbitrary prime. Let 𝜌 p ( m ) denote the exponent of the highest power of a prime p which divides m . Recently, the author obtained lower bounds for 𝜌 p ( y n ) based on recurrence relations of order p and p m 1. The cases with tight bounds have also been characterized. In this paper, we prove that 𝜌 p ( y np ( p , a ))= n for 𝜌 p ( a +1)=1, p S 3. We obtain alternative difference equations of order p 2 for y n and order p for the p -sected sequence y np by a generating function based method. We also extend general divisibility results relying on the arithmetic properties of the coefficients and initial values.     

Nonstanard finite difference scheme for a scalar reaction-convection PDE

This paper is dedicated to my friend and colleague, professor Greey Ladas, in celebration of his sixtieth birthday. May his outstanding contributions to mathematics continue for many years.

We construct a nonstrandard finite difference shceme for a scalar reaction-convection partial differntial equation (PDE) having a symmetric, nonlinear cubic source term in the dependen variable. Is mathematical properies are studied and compared with those of the original PDE. We also construct the exact scheme for this equations.     

Asymptotic Diagonalization and Normalization of Linear q -Difference Equations

Asymptotic diagonalizations of linear differential equations are studied by several authors. The problems for linear difference equations are investigated recently by Bodine and Sacker. In their work, the full spectrum condition plays essential role. Here we consider a related problem for q-difference equations, |q| < 1, which do not satisfy the full spectrum condition. Our tool is the Arnold normal form for matrix.     

Almost periodic solutions for a class of non-instantaneous impulsive differential equations

Abstract

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations.     

Extrapolation of the Stochastic Theta Numerical Method for Stochastic Differential Equations

ABSTRACT

The stochastic theta method is a family of implicit Euler methods for approximating solutions to Itô stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form to apply Richardson extrapolation. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation.