Nonlinear Differential-difference Equations Research Articles

New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology

We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential—difference equations. The proposed method is based on the Laplace transform with the homotopy analysis method (HAM). This method is a powerful tool for solving a large amount of problems. This technique provides a series of functions which may converge to the exact solution of the problem. A good agreement between the obtained solution and some well-known results is obtained.

Some results related to complex linear and nonlinear difference equations

In this paper, we investigate the complex oscillation problems of meromorphic solutions to some linear difference equations with meromorphic coefficients, and obtain some results about the relationships between the exponent of convergence of zeros, poles and the order of growth of meromorphic solutions to complex linear difference equations. We also study the existence of solution of certain types of nonlinear differential-difference equations, and partially answer a conjecture concerning the above problem posed by Yang and Laine (C.C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86(1) (2010), pp. 10–14).

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.

A reliable algorithm for solving discontinued problems arising in nanotechnology

In this paper, a reliable algorithm based on new homotopy perturbation transform method (HPTM) is proposed to solve a nonlinear differential-difference equation arising in nanotechnology. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. The HPTM is a combined form of Laplace transform, homotopy perturbation method and He’s polynomials. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The numerical solutions show that the proposed method is very efficient and computationally attractive. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems.

Modeling the bursting effect in neuron systems

We propose a new method for modeling the well-known phenomenon of “bursting behavior” in neuron systems by invoking delay equations. Namely, we consider a singularly perturbed nonlinear difference-differential equation with two delays describing the functioning of an isolated neuron. Under a suitable choice of parameters, we establish the existence of a stable periodic motion with any prescribed number of spikes on a closed time interval equal to the period length.

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.

Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time

We consider heat and diffusion equations with finite relaxation time which ensure a finite speed of propagation of disturbances. We use the Cattaneo–Vernotte model for the heat flux and obtain a number of exact solutions to the corresponding linear differential-difference heat equation. We also give exact solutions to two one-dimensional Stokes problem for a differential-difference mass/heat transfer equation (without a source and with a linear source) with a periodic boundary condition.We describe a number of exact solutions to non-linear differential-difference heat equations of the formT¯t=div[f(T)∇T]+g(T¯),T¯=T(x,t+τ),where τ is the relaxation time. In addition, we obtain some exact solutions to non-linear systems of two coupled reaction–diffusion equations with finite relaxation time and present several exact solutions of non-linear reaction–diffusion equations with time-varying delay of the formut=kuxx+F(u,w),w=u(x,t−τ),where τ=τ(t).All equations in question contain arbitrary functions or free parameters. Their solutions can be used to solve certain problems and test numerical methods for non-linear partial differential-difference equations (delay partial differential equations).

Chaotic dynamics of the fractional order nonlinear system with time delay

This paper presents the fractional order model of a nonlinear autonomous continuous-time difference-differential equation with only one variable. Numerical simulation results of the fractional order model demonstrate the existence of chaos when system order $$q\ge 0.2$$ . Values of the delay time $$\tau $$ in which chaotic behavior is observed at system order $$q$$ are quantitatively defined using the largest Lyapunov exponents obtained from the output time series.

Reliable analysis for time-fractional nonlinear differential difference equations

AbstractIn this study, we used HPM to determine the approximate analytical solution of nonlinear differential difference equations of fractional time derivative. By using initial conditions, the explicit solutions of the coupled nonlinear differential difference equations have been derived which demonstrate the effectiveness, potentiality and validity of the method in reality. The present method is very effective and powerful to determine the solution of system of non-linear DDE. The numerical calculations are carried out when the initial condition in the form of hyperbolic functions and the results are shown through the graphs.

On Some Methods for Study of Stochastic Hereditary Systems

Different types of stochastic retarded systems could be considered on the base of our schemes of phase space extension. This scheme is applicable for investigation of linear and nonlinear differential difference equations with single and multiple constant delays, differential equations with variable delays, neutral delay differential equations etc. In addition, a combination of the scheme and the Monte Carlo method can be exploited for study sophisticated objects. The main topic of research in this paper is the problem of computation of the first moment functions for the phase vectors defining states for linear stochastic systems of differential equations with the special form of variable delays, notably with piecewise constant delays, and integro-differential equations.

Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

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.

GDTM-Padé technique for the non-linear differential-difference equation

This paper focuses on applying the GDTM-Pad? technique to solve the non-linear differential-difference equation. The bell-shaped solitary wave solution of Belov-Chaltikian lattice equation is considered. Comparison between the approximate solutions and the exact ones shows that this technique is an efficient and attractive method for solving the differential-difference equations.

Automated derivation of the conservation laws for nonlinear differential-difference equations

Based on Wu’s elimination method and “divide-and-conquer” strategy, the undetermined coefficient algorithm to construct polynomial form conservation laws for nonlinear differential-difference equations (DDEs) is improved. Furthermore, a Maple package named CLawDDEs, which can entirely automatically derive polynomial form conservation laws of nonlinear DDEs is presented. The effectiveness of CLawDDEs is demonstrated by application to different kinds of examples.

Exact and explicit solutions for the discrete Burgers equation by means of the Exp-function method

In recent years, various direct methods were proposed to find exact solutions not only for nonlinear partial differential equations but also for nonlinear differential-difference equations (NDDEs). In this paper, the Exp-function method was extended to the discrete Burgers equation for searching traveling waves in closed forms. It was shown that the equation supported discrete solitary wave solutions. Our results, as test problems, might be of great importance for those who study in the field of numerical analysis. It can be concluded that the Exp-function approach provides a direct and efficient way to construct generic exponential wave solutions to NDDEs. Key words: Exp-function method, discrete Burgers equation, nonlinear differential-difference equation.

Preliminary group classification of the nonlinear differential–difference equations

Preliminary group classification of the nonlinear differential–difference equations of the form ün=Fn(t,un−1,un,un+1) is considered by using the classical infinitesimal Lie method and the theory of classification of abstract low-dimensional Lie algebras. Restrict to Lie point symmetries, and assume that the Lie algebra of the symmetry group is realized by a particular vector field. It is proved that there is only one equation admitting three-dimensional simple Lie algebras. What is more, all the inequivalent equations admitting semi-simple Lie algebras are nothing but the one. Also there exist inequivalent equations admitting low-dimensional solvable Lie algebras and semi-direct sums of semi-simple and solvable Lie algebras.

Integrable nonlinear differential-difference hierarchy and Darboux transformation

In this paper, with a free function embedded into a discrete zero-curvature equation, an integrable nonlinear differential-difference hierarchy is derived via the extension of original hierarchy with symbolic computation. Based on the Lax pair, infinitely many conservation laws and Darboux transformations are constructed for the first nonlinear differential-difference equations in the hierarchy. As an application of the Darboux transformation, some explicit solutions of those sample equations are given.

A Numerical Patching Technique for Singularly Perturbed Nonlinear Differential-Difference Equations with a Negative Shift

In this paper, we present a numerical patching technique for solving singularly perturbed nonlinear differen- tial-difference equation with a small negative shift. The nonlinear problem is converted into a sequence of linear problems by quasilinearization process. After linearization, it is divided into two problems, namely inner region problem and outer region problem. The boundary condition at the cutting point is obtained from the theory of singular perturbations. Using stretching transformation, a modified inner region problem is constructed and is solved by using the upwind finite difference scheme. The outer region problem is solved by a Taylor polynomial approach. We combine the solutions of both problems to obtain an approximate solution of the original problem. The proposed method is iterative on the cutting point. The process is repeated for various choices of the cutting point, until the solution profiles stabilize. Some numerical examples have been solved to demonstrate the applicability of the method. The method is analyzed for stability and convergence.

Discrete autowaves in systems of delay differential-difference equations in ecology

We propose a theory of relaxation oscillations for a nonlinear scalar delay differential-difference equation that represents a modification of the well-known Hutchinson equation in ecology. In particular, we establish that a one-dimensional chain of diffusively coupled equations of this type exhibits the well-known buffer phenomenon. Namely, under an increase in the number of links in the chain and a consistent decrease in the coupling constant, the number of coexisting stable periodic motions indefinitely increases.

Discrete autowaves in neural systems

A singularly perturbed scalar nonlinear differential-difference equation with two delays is considered that is a mathematical model of an isolated neuron. It is shown that a one-dimensional chain of diffusively coupled oscillators of this type exhibits the well-known buffer phenomenon. Specifically, as the number of chain links increases consistently with decreasing diffusivity, the number of coexisting stable periodic motions in the chain grows indefinitely.