Entire functions sharing one polynomial with their derivatives

In this paper, we study the growth of solutions of ak-th order linear differential equation and that of a k + 1-th order linear differential equation. From this we affirmatively answer a uniqueness question concerning a conjecture given by Brück in 1996 under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its derivative sharing a finite nonzero complex number CM. The results in this paper also improve some known results. Some examples are provided to show that the results in this paper are best possible.     

On the uniqueness of an entire function sharing a small entire function with some linear differential polynomial

We prove a theorem on the growth of nonconstant solutions of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its linear differential polynomial share a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.     

整函数及其微分多项式分担一个多项式

将Bru¨ck猜想目前得到的几个结论进行了推广,研究了整函数及其微分多项式分担的一个多项式时的问题,并且得到了一个与之相关的复微分方程的解的性质.另外,还得到了一个定理,这个定理改进了一些已知的结果.     

The uniqueness of an entire function sharing a small entire function with its derivatives

In this paper, we prove a theorem on the growth of a solution of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.     

Analytical solution of the linear fractional differential equation by Adomian decomposition method

In this paper, we consider the n-term linear fractional-order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by Adomian decomposition method. With the equivalent transmutation, we show that the solution by Adomian decomposition method is the same as the solution by the Green's function. Finally, we illustrate our result with some examples.     

脉冲中立型时滞微分方程的正解的存在性

本文证明了线性脉冲中立型微分方程 [y(t)+p(t)y(t-τ)-r(t)y(t-ρ)]′+q(t)y(t-σ)=0,t≥0,t≠t_k, y(t_k~+)-y(t_k)=b_ky(t_k),k=1,2,…正解的存在性等价于一类非脉冲中立型方程正解的存在性,应用这一结果,建立了此类线性脉冲中立型微分方程正解存在性的若干充分条件。     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …     

一个微分方程的解及其应用

本文利用Gundersen的方法研究了一个线性微分方程的解并证明此方程的每一个整函数解的级必为无穷,推广了Gundersen和杨连中的若干结果。作为应用,我们还研究了与导数具有公共不动点的整函数.     

Exact master equations describing reduced dynamics of the Wigner function

Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations. In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this, we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 203–219, 2005.     

二阶时滞微分方程非振动性质在脉冲扰动下的不变性

本文建立了一类二阶脉冲时滞微分方程解的一个整体存在唯一性定理,并讨论了脉冲扰动对脉冲线性时滞微分程非振动性质的影响,获得了脉冲扰动对时滞微分方程非振动性质没有影响的一般性脉冲条件,推广了最近某些文献中的结论.     

A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k , where the order k is even unnecessary to be equal to the order n . In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.     

On uniqueness of an entire function and its derivatives

In this paper, we study the growth of solutions of a first order linear differential equation. From this we verify that a conjecture given by Brück is true under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its first derivative sharing a nonzero constant CM. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible. Project supported by the NSFC (NO. A0324617) and the RFDP (NO. 20060422049). Received: 20 April 2006 Revised: 4 February 2007     

临界状态下一阶时滞微分方程的线性化振动性

本文首先在临界状态下建立了一阶非线性非自治时滞微分方程ｘ＇（ｔ）＋＝１ｐｉｘ（ｔ－Ｔｉ）＋ｆ（ｔ,ｘ（ｔ－σ１（ｔ））…,ｘ（ｔ－σｎ（ｔ））＝０与一个相关的二阶常微分方程振动性等价定理,进而给出了一阶非线性自治微分方程与相应的线性方程振动性等价的充分条件,从而较好地回答了张炳根在文［２］中提出的一个公开问题．     

关于Borg定理

Borg定理是判定周期系数二阶线性微分方程稳定的一个重要定理,这个定理在一定意义下是不可改进的.本文利用判别式的一个新形式,在弱的限制下,得到判定周期系数二阶线性微分方程稳定的定理,所得结论改进了Borg定理的判定结果.     

Spline collocation methods for linear multi-term fractional differential equations

Some regularity properties of the solution of linear multi-term fractional differential equations are derived. Based on these properties, the numerical solution of such equations by piecewise polynomial collocation methods is discussed. The results obtained in this paper extend the results of Pedas and Tamme (2011) [15] where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer. In the present paper, we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using Caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by some numerical examples.     

Spline collocation methods for linear multi-term fractional differential equations

Some regularity properties of the solution of linear multi-term fractional differential equations are derived. Based on these properties, the numerical solution of such equations by piecewise polynomial collocation methods is discussed. The results obtained in this paper extend the results of Pedas and Tamme (2011) [15] where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer. In the present paper, we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using Caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by some numerical examples.     

Trichotomies for abstract semilinear differential equations

In this paper we present some results concerning the existence and uniqueness of mild solutions to certain abstract semilinear differential equations and the asymptotic behavior of these solutions. The basic techniques used are the iterative method and the fixed point theory for differential equations in Banach space. However, the most pleasant here is that it can be applied to nonlinear equations without assuming the eigenvalues of the differential operator in the linear parts of the differential equation has non-zero real part.     

Perturbation method for nonlocal impulsive evolution equations

This paper deals with the existence of mild solutions for a class of semilinear nonlocal impulsive evolution equations in ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear nonlocal impulsive evolution equation is established. With the aid of the theorem, the existence of mild solutions for nonlinear nonlocal impulsive evolution equation is obtained by using perturbation method and monotone iterative technique. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. Moreover, we present two examples to illustrate the feasibility of our abstract results.     

Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.