Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay

In this work, we give a variation of constants formula for partial functional differential equations with infinite delay. We assume that the linear part is not necessarily densely defined and its resolvent operator satisfies the Hille-Yosida condition. We establish a reduction of the problem to a finite-dimensional space which allows us to prove the existence of almost periodic solutions.     

The uniform asymptotic stability of certain neutral differential-difference equations

In this paper we give a necessary and sufficient condition for a general class of neutral differential-difference equations to be exponentially stable. This condition is expressed in terms of certain bilinear functionals which are the equivalent of quadratic Liapunov functions for finite-dimensional systems.     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

Pseudo almost periodic solutions of infinite class for some functional differential equations

The aim of this work is to investigate the existence and uniqueness of pseudo almost periodic solutions for some neutral partial functional differential equations in a Banach space when the delay is distributed using the variation of constants formula and the spectral decomposition of the phase space developed in Adimy et al. [M. Adimy, K. Ezzinbi, and A. Ouhinou, Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 317(2) (2006), pp. 668–689]. Here, we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part is assumed to be pseudo almost periodic with respect to the first argument and Lipschitz continuous with respect to the second argument.     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.     

Stepanov ergodic perturbations for some neutral partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of μ?pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov μ?pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main results, we give applications to a neutral model arising in physical systems, as well as an application to heat equations with discrete and continuous delay. Copyright © 2015 John Wiley & Sons, Ltd.     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

A discrete generalization of the extended simplest equation method

We modified the so-called extended simplest equation method to obtain discrete traveling wave solutions for nonlinear differential-difference equations. The Wadati lattice equation is chosen to illustrate the method in detail. Further discrete soliton/periodic solutions with more arbitrary parameters, as well as discrete rational solutions, are revealed. We note that using our approach one can also find in principal highly accurate exact discrete solutions for other lattice equations arising in the applied sciences.     

Jacobian elliptic function method for nonlinear differential-difference equations

An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrödinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.     

Variation of constants formula and reduction principle for a class of partial functional differential equations with infinite delay

In this work, the dynamic behavior of solutions is investigated for a class of partial functional differential equations with infinite delay. We suppose that the undelayed homogeneous part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator. Firstly, a variation of constants formula is obtained in the corresponding α-norm space, which is mainly used to establish a reduction principle of complexity of the considered equation. The reduction principle proves that the dynamics of the considered equation is governed by an ordinary differential equation in finite dimensional space. As an application, we investigate the existence of periodic, almost periodic and almost automorphic solutions for the original equation.     

Homoclinic Points for a Singularly Perturbed System of Differential Equations with Delay

We obtain a representation of the integral manifold of a system of singularly perturbed differential-difference equations with periodic right-hand side. We show that, under certain conditions imposed on the right-hand side, the Poincaré map for the perturbed system has a transversal homoclinic point.     

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift-Hohenberg equation and reaction-diffusion systems of activator-inhibitor type.     

The exponential function rational expansion method and exact solutions to nonlinear lattice equations system

We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem.     

SOME PROBLEMS IN RADIATION TRANSPORT FLUID MECHANICS AND QUANTUM FLUID MECHANICS

We introduce the radiation transport equations, the radiation fluid mechanics equations and the fluid mechanics equations with quantum effects. We obtain the unique global weak solution for the radiation transport fluid mechanics equations under certain initial and boundary values. In addition, we also obtain the periodic region problem of the compressible N-S equation with quantum effect has weak solutions under some conditions.     

NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS OF NEUTRAL LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS：A METHOD BY FOURIER SERIESES

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The solutions of Toda lattice and Volterra lattice

This paper presents a method to directly construct explicit exact solutions to nonlinear differential-difference equations. One applies this approach to solve Volterra lattice and Toda lattice and obtain their some special solutions which contain soliton solutions and periodic solutions.     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

Global weak solution for a periodic generalized Hunter–Saxton equation

In this paper, we consider a periodic generalized Hunter–Saxton equation. We obtain the existence of global weak solutions to the equation. First, we give the well-posedness result of the viscous approximate equations and establish the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the equation.