Periodic boundary value problems for impulsive fuzzy differential equations

The peculiarity of the Hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations. In this work, we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method. Fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic ‘linear’ differential problems is also provided.     

Periodic Solutions for a Class of Second-Order Ordinary Differential Equations

This paper is devoted to the study of periodic solutions for a class of second-order ordinary differential equations by utilizing a technique for obtaining solutions to free problems in the calculus of variations originating in the work of Carathéodory (Ref. 1, 1935). The key of this technique is to find some suitable transformation which transfers the periodic solution problem to an equivalent variational problem in which the minimizer is more easily determined. Some applications are presented to illustrate the utility of this technique.Supported by NSFC Grant 10401013, 985 Project of Jilin University and Graduate Innovation Lab of Jilin University. The authors thank Professor D. A. Carlson and the anonymous referees for valuable suggestions and helpful comments.     

Chaotic Behavior in Slowly Varying Systems of Nonlinear Differential Equations

A technique is developed to find parameter regions of chaotic behavior in certain systems of nonlinear differential equations with slowly varying periodic coefficients. The technique combines previous results on how to find branches of periodic solutions which terminate with a homoclinic orbit and results on how to find chaotic trajectories in the neighborhood of homoclinic trajectories of the autonomous system. The technique is applied to the continuous stirred tank reaction A → B, for which it is shown that a slowly varying periodic flow rate can yield aperiodic temperature fluctuations.     

单调迭代法与一阶脉冲泛函微分方程周期边值问题

本文利用单调迭代法与上下解方法,研究了一阶脉冲泛函做分方程周期边值问题最大解与最小解的存在性.     

Solving time‐periodic fractional diffusion equations via diagonalization technique and multigrid

This paper addresses numerical computation of time‐periodic diffusion equations with fractional Laplacian. Time‐periodic differential equations present fundamental challenges for numerical computation because we have to consider all the discrete solutions once in all instead of one by one. An idea based on the diagonalization technique is proposed, which yields a direct parallel‐in‐time computation for all the discrete solutions. The major computation cost is therefore reduced to solve a series of independent linear algebraic systems with complex coefficients, for which we apply a multigrid method using the damped Richardson iteration as the smoother. Such a linear solver possesses mesh‐independent convergence factor, and we make an optimization for the damping parameter to minimize such a constant convergence factor. Numerical results are provided to support our theoretical analysis.     

Periodic boundary value problem for first-order impulsive ordinary differential equations

This paper investigates the existence of minimal and maximal solutions of the periodic boundary value problem for first-order impulsive differential equations by establishing two comparison results and using the method of upper and lower solutions and the monotone iterative technique.     

Bifurcations of periodic solutions of delay differential equations

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times.     

Periodic boundary value problem for first-order impulsive functional differential equations

In this paper, by means of the method of upper and lower solutions and the monotone iterative technique, the existence of minimal and maximal solutions of the periodic boundary value problem for first-order impulsive functional differential equations is considered.     

Differential equations without uniqueness and classical topological dynamics

In this paper we present a point of view which allows one to interpret the solutions of a non-autonomous differential equation as a classical dynamical system, without assuming uniqueness of solutions of the initial value problem. In addition we are able to construct a global flow without assuming the global existence of solutions of the given differential equations. This point of view seems appropriate in the sense that many applications of the results of classical topological dynamics to the study of the solutions of differential equations can now be performed, in a straight-forward manner, without the uniqueness assumption. We shall illustrate this claim with one important application concerning the existence of periodic or almost 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。     

Monotone Method for Second Order Periodic Boundary Value Problems and Periodic Solutions of Delay Difference Equations

In this article, we employ monotone iterative technique to study the existence of solutions for second order periodic boundary value problem and periodic solutions of delay difference equations.     

Existence of Periodic Solutions in Impulsive Differential Equations

In this paper, we are concerned with the problem of existence of periodic solutions for a class of second order impulsive differential equations. By Poincaré-Bohl theorem, we give several criteria to guarantee that the impulsive differential equation has periodic solutions under assumptions that the nonlinear term satisfies the linear growth conditions. Two specific examples are presented to illustrate the obtained results.     

On some properties of degenerate linear systems

For a linear system of ordinary differential equations with degenerate matrix of derivatives, we find conditions of reducibility to the central canonical form. We also establish the structure of the general solution and conditions of solvability of the Cauchy problem, and study the problem of periodic solutions. Nezhin Pedagogic Institute, Nezhin. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1278–1296, September, 1997.     

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

The necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional differential equations

In this paper, using Fourier series, we study the problem of the existence of periodic solutions of a type of periodic neutral differential difference system. Some necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional equation system are obtained, and at the same time, we present a method with formula shows how to find the periodic solutions.     

The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations

This paper uses the method of upper and lower solutions and the monotone iterative technique to investigate the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive functional differential equations. Received June 15, 1998, Revised April 29, 2000, Accepted July 18, 2000     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Analytical treatment and convergence of the Adomian decomposition method for a system of coupled damped wave equations

An abstract result is proved for the convergence of Adomian decomposition method for partial differential equations that model evolutionary systems. Moreover, we prove that this decomposition scheme applied to a system of wave equations coupled in parallel with homogeneous Dirichlet boundary conditions, is convergent in a suitable Hilbert space. Furthermore, this technique is utilized to find closed-form solutions for the problem under consideration.