非线性微分方程的正周期解

§ 1　 IntroductionIn[1 ] ,Saker and Agarwal studied the existence and uniqueness of positive periodicsolutions of the nonlinear differential equationN′(t) =-δ(t) N(t) + p(t) N(t) e- a N(t) ,(1 )whereδ(t) and p(t) are positive T-periodic functions.They proved that if p* >δ* ,then(1 ) has a unique T-periodic positive solution,wherep* =min0≤ t≤ Tp(t) ,δ* =max0≤ t≤ Tδ(t) .　　In view ofthe papermentioned above,whatcan be said aboutequation(1 ) when p* ≤δ* ?In this paper,we conside…     

Applying carvalho's method to find periodic solutions of difference equations

In this paper we apply the method initially developed in [1] for differential-difference equations, to the case of difference equations, in order to find 2 and 3-periodic solutions of some equations that often appear in the literatures as are for instance the case of Applications 2,5 which are examples of population growth models, and Application 4, which is a standard example of nonlinear higher order scalar difference equation depending on two parameters (see, Kocik and Ladas [3]).     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.     

Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation $u^{″}\left(t\right)+a\left(t\right)u\left(t\right)=f(t,u\left(t\right))\text{,}t\in R\text{,}$ where $a:R\to [0,+\infty )$ is an $\omega$-periodic continuous function with $a\left(t\right)⁄\equiv 0$, $f:R\times [0,+\infty )\to [0,+\infty )$ is continuous and $f(\cdot ,u):R\to [0,+\infty )$ is also an $\omega$-periodic function for each $u\in [0,+\infty )$. Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.     

Three periodic solutions of nonlinear neutral functional differential equations

In this paper, a type of nonlinear neutral functional differential equations are considered. Some results on the existence of three positive periodic solutions are obtained by using the Leggett–Williams fixed point theorem.     

Solution of nonlinear fractional differential equations using homotopy analysis method

In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A one-step optimal homotopy analysis method for nonlinear differential equations

In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.     

On exact solutions to partial differential equations by the modified homotopy perturbation method

Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the effciency of the proposed method.     

Analysis of nonlinear fractional partial differential equations with the homotopy analysis method

In this paper, the time fractional partial differential equations are investigated by means of the homotopy analysis method. This technique is extended to study the partial differential equations of fractal order for the first time. The accurate series solutions are obtained. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional partial differential equations.     

Numerical solutions of partial differential equations by discrete homotopy analysis method

This paper introduces a discrete homotopy analysis method (DHAM) to obtain approximate solutions of linear or nonlinear partial differential equations (PDEs). The DHAM can take the many advantages of the continuous homotopy analysis method. The proposed DHAM also contains the auxiliary parameter ?, which provides a simple way to adjust and control the convergence region of solution series. The convergence of the DHAM is proved under some reasonable hypotheses, which provide the theoretical basis of the DHAM for solving nonlinear problems. Several examples, including a simple diffusion equation and two-dimensional Burgers’ equations, are given to investigate the features of the DHAM. The numerical results obtained by this method have been compared with the exact solutions. It is shown that they are in good agreement with each other.     

Optimal homotopy analysis method for nonlinear differential equations in the boundary layer

Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. There are several approaches to determine the optimal values of these parameters, which can be divided into two categories, i.e. global optimization approach and step-by-step optimization approach. In the global optimization approach, all the parameters are optimized simultaneously at the last order of approximation. However, this process leads to a system of coupled, nonlinear algebraic equations in multiple variables which are very difficult to solve. In the step-by-step approach, the optimal values of these parameters are determined sequentially, that is, they are determined one by one at different orders of approximation. In this way, the computational efficiency is significantly improved, especially when high order of approximation is needed. In this paper, we provide extensive examples arising in similarity and non-similarity boundary layer theory to investigate the performance of these approaches. The results reveal that with the step-by-step approach, convergent solutions of high order of approximation can be obtained within much less CPU time, compared with the global approach and the traditional HAM.     

Newton's method and periodic solutions of nonlinear wave equations

We prove the existence of periodic solutions of the nonlinear wave equation satisfying either Dirichlet or periodic boundary conditions on the interval [O, π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.     

Existence of positive periodic solutions of nonlinear first-order delayed differential equations

We consider the existence of positive ω-periodic solutions for the equation

u^′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),