Solutions Of Nonlinear Differential-difference Equations Research Articles

Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions

Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions

On Meromorphic Solutions of Non-linear Differential-Difference Equations

In this paper, we investigate the non-existence of transcendental entire solutions for non-linear differential-difference equations of the forms fn(z)+Q(z,f)=β1eα1z+β2eα2z+⋯+βseαsz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f^{n}(z)+Q(z,f)=\\beta _{1}e^{\\alpha _{1}z}+\\beta _{2}e^{\\alpha _{2}z}+\\cdots +\\beta _{s}e^{\\alpha _{s}z} \\end{aligned}$$\\end{document}and fn(z)f(k)(z)+Ld(z,f)=∑i=1spi(z)eαi(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f^{n}(z)f^{(k)}(z)+L_d(z,f)=\\sum ^{s}_{i=1}p_i(z)e^{\\alpha _i{(z)}}, \\end{aligned}$$\\end{document}where n, s are positive integers, nge s+2,Q(z, f) is a differential-difference polynomial in f of degree d.

Exponential polynomialsas solutions of non-linear differential-difference equations

Exponential polynomials, an important subclass of finite order entire functions, as solutions of differential or difference or differential-difference equations are considered in [5, 10, 19, 20]. The critical domains of zeros and the quotients of exponential polynomials are considered in [6]. In this paper, we proceed to consider the exponential polynomials as solutions of some general complex differential-difference equations and extend existence results.

Exponential Polynomials as Solutions of Nonlinear Differential-Difference Equations

Exponential Polynomials as Solutions of Nonlinear Differential-Difference Equations

The growth of entire solutions of certain nonlinear differential-difference equations

<abstract><p>This paper is concerned with entire solutions of nonlinear differential-difference equations. We will characterize the growth of entire solutions for two classes of nonlinear differential-difference equations. Our results will contribute to the generalization and completion of some results obtained recently. The results are explained by various examples and remarks.</p></abstract>

Exponential Polynomials and Nonlinear Differential-Difference Equations

In this paper, we study finite-order entire solutions of nonlinear differential-difference equations and solve a conjecture proposed by Chen, Gao, and Zhang when the solution is an exponential polynomial. We also find that any exponential polynomial solution of a nonlinear difference equation should have special forms.

Entire solutions of nonlinear differential-difference equations.

In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift f(z+c), which is a generalization of a result of Liu.

Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types

Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous. Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations, the majority of the results deal with polynomial types. Limited research has been reported regarding such equations of rational type. In this paper we present an adaptation of the (G′/G)-expansion method to solve nonlinear rational differential-difference equations. The procedure is demonstrated using two distinct equations. Our approach allows one to construct three types of exact traveling wave solutions (hyperbolic, trigonometric, and rational) by means of the simplified form of the auxiliary equation method with reduced parameters. Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.

Analytical Approximate Solutions for Nonlinear Differential-Difference Equations Arising in Nanotechnology

Analytical Approximate Solutions for Nonlinear Differential-Difference Equations Arising in Nanotechnology

Conditions for almost periodicity of bounded solutions of non-linear differential-difference equations

We consider non-linear almost periodic differential- difference equations having solutions with precompact sets of values. We obtain conditions for such solutions to be almost periodic.

Kink-Type Solutions of the MKdV Lattice Equation with an Arbitrary Function

In this paper, the exp-function method is improved for constructing exact solutions of nonlinear differential-difference equations with variable coefficients. To illustrate the validity and advantages of the improved method, the mKdV lattice equation with an arbitrary function is considered. As a result, kink-type solutions are obtained which possess rich spatial structures. It is shown that the improved exp-function method can be applied to some other nonlinear differential-difference equations with variable coefficients.

Exact solutions of non-linear differential-difference equations of a viscous fluid with finite relaxation time

We have found a number exact of solutions to non-linear differential-difference equations of a viscous incompressible fluid with finite relaxation time. The results are used to solve a few hydrodynamic problems; in particular, we describe a one-dimensional flow arising from longitudinal periodic oscillations of a rigid plane and a two-dimensional periodic flow with pressure gradient near a fixed porous plate. We discuss issues of hydrodynamic instability of solutions as well as ways to enhance differential-difference fluid and related heat and diffusion models. The modified relaxation fluid model presented may theoretically explain one of the causes of the onset of turbulence. We also give a few exact solutions to the Racke–Saal non-linear hyperbolic fluid equations.

Extended rational expansion method for differential-difference equation

In this paper, the rational formal expansion method is extended to construct a series of soliton-like and period-form solutions for nonlinear differential-difference equations. The efficiency of the method has been demonstrated on Volterra Lattice. Compared with most existing methods, the proposed method not only recover some known solutions, but also find some new and more general solutions.

A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

Application of Rational Expansion Method for Differential-Difference Equation

In this paper, we applied the rational formal expansion method to construct a series of soliton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some known solutions, but also finds some new and more general solutions. The efficiency of the method can be demonstrated on Toda Lattice and Ablowitz—Ladik Lattice.

New exact solutions of nonlinear differential-difference equations with symbolic computation

In this paper, the Toda equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity via the discrete \( \left( {\frac{{G'}}{G}} \right) \)-expansion method are researched. As a result, with the aid of the symbolic computation, new hyperbolic function solution and trigonometric function solution with parameters of the Toda equation are obtained. At the same time, new envelop hyperbolic function solution and envelop trigonometric function solution with parameters of the discrete nonlinear Schrodinger equation with a saturable nonlinearity are obtained. This method can be applied to other nonlinear differential-difference equations in mathematical physics.

Exp-Function Method for N-Soliton Solutions of Nonlinear Differential-Difference Equations

In this paper, the exp-function method is generalized to construct N-soliton solutions of nonlinear differential-difference equations. With the aid of symbolic computation, we choose the Toda lattice to illustrate the validity and advantages of the generalized work. As a result, 1-soliton, 2-soliton, and 3-soliton solutions are obtained, from which the uniform formula of N-soliton solutions is derived. It is shown that the exp-function method may provide us with a straightforward and effective mathematical tool for generating N-soliton solutions of nonlinear differential-difference equations in mathematical physics.

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.

Discrete Jacobi elliptic function expansion method for nonlinear differential-difference equations

In this paper, an improved algorithm is devised to derive exact travelling wave solutions of nonlinear differential-difference equations (DDEs) by means of Jacobi elliptic functions. With the aid of symbolic computation, we choose the integrable discrete nonlinear Schrödinger equation to illustrate the validity and advantages of the method. As a result, new and more general Jacobi elliptic function solutions are obtained, from which hyperbolic function solutions and trigonometric function solutions are derived when the modulus m→1 and 0. It is shown that the proposed method provides a more effective mathematical tool for nonlinear DDEs in mathematical physics.

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.