EXISTENCE OF HOMOCLINIC ORBITS FOR A CLASS OF FIRST-ORDER DIFFERENTIAL DIFFERENCE EQUATIONS

In this article we consider via critical point theory the existence of homoclinic orbits of the first-order differential difference equation z（t）＋B（t）z（t）＋f（t,z（t＋τ）,z（t）,z（t-τ））=0.

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.

Almost Periodic Solutioiis of Neutral Differential Difference Equations with Piecewise Constant Arguments

In this paper,we study the existence of almost periodic solutions of neutral differential difference equations with piecewise constant arguments via difference equation methods.

New exact solutions to some difference differential equations

In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.

ASYMPTOTIC SOLUTION TO SINGULARLY PERTURBED NEUTRAL DIFFERENTIAL DIFFERENCE EQUATIONS

The solvability for a kind of singularly perturbed problem of nonlinear neutral differential difference system is considered. Using the boundary layer corrective method, the formal asymptotic solution is constructed. And applying the theory of fixed point, the uniform validity of the asymptotic expansions for solution is proved. Finally, an example is given to validate the results of the problems.

ON THE OSCILLATION OF NEUTRAL DIFFERENTIAL DIFFERENCE EQUATIONS WITH "INTEGRALLY SMALL" COEFFICIENTS

Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results on the oscillation of Eq.() without the usual requirement that We also give some interesting open problems on this topic.

Existence of Many and Infinitely Many Periodic Solutions for Some Types of Differential Delay Equations

J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).

A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.

The Differential Transform

One of the methods of mathematical analysis in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator d/dt and solved a number of applied problems. However, he did not give operational calculus a mathematical basis;this was done with the aid of the Laplace transform;J. Mikusiński (1953) put operational calculus into algebraic form, using the concept of a function ring [1]. Thereupon I’m suggesting here the use of the integration operator dt to make an extension for the systematic use of operational calculus. Throughout designing and analyzing a control system, we need to treat the functionality of the system with respect to time. The reaction of the system instructs us how to stable it by amplifiers and feedbacks [2], thereafter the Differential Transform is a good tool for doing this, and it’s a technique to frustrate difficulties we may bump into, also it has the methods to find the immediate reaction of the system with respect to infinitesimal (tiny) time which spares us from the hard work in finding the time dependent function, this could be done by producing finite series.

THE ANALYSIS OF EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR SECOND ORDER LINEAR SINGULAR DIFFERENTIAL DIFFERENCE EQUATION WITH DELAY

This paper discusses all cases of second order linear singular defferential difference equations with delay and different coefficients, and prensents the conditionsfor existence and uniqueness of solutions nearly in all cases.

Infinite horizon LQR for systems with multiple delays in a single input channel

This paper is concerned with the linear quadratic regulation （LQR） problem for both linear discrete-time systems and linear continuous-time systems with multiple delays in a single input channel. Our solution is given in terms of the solution to a two-dimensional Riccati difference equation for the discrete-time case and a Riccati partial differential equation for the continuous-time case. The conditions for convergence and stability are provided.

THE EXISTENCE OF POSITIVE SOLUTIONS OF THE NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS IN A BANACH SPACE

In this paper, the author obtains the existence of positive solutions of nonlinear neutral differential difference equations in a Banach space by means of the fixed point theorems.

NECESSARY AND SUFFICIENT CONDITION FOR OSCILLATION OF A CLASS OF HIGHERORDER NEUTRAL EQUATIONS

Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > 0, σ> 0 and q are all constants.

One-dimensional monotone nonautonomous dynamical systems

This work is devoted to the study of the dynamics of one-dimensional monotone nonautonomous(cocycle) dynamical systems. A description of the structures of their invariant sets, omega limit sets,Bohr/Levitan almost periodic and almost automorphic motions, global attractors, and pinched and minimalsets is given. An application of our general results is given to scalar differential and difference equations.