About stability of a difference analogue of a nonlinear integro-differential equation of convolution type

A nonlinear integro-differential equation of convolution type with order of nonlinearity more than one and a stable trivial solution is considered. The integral in this equation has an exponential kernel and polynomial integrand. The difference analogue of the equation considered is constructed in the form of a difference equation with continuous time and it is shown that this difference analogue preserves the properties of stability of his original.     

On stability and bifurcation of periodic solutions of delay differential equations

The purpose of this paper is to study a class of delay differential equations with two delays. first, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the correlocal stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.     

On the Equation of Motion of a Thin Layer of Uniform Vorticity

A higher order extension to Moore's equation governing the evolution of a thin layer of uniform vorticity in two dimensions is obtained. The equation, in fact, governs the motion of the center line of the layer and is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line. It extends Birkoff's equation for a vortex sheet. The equation is used to examine the growth of disturbances on a straight, steady layer of uniform vorticity. The growth rate for long waves is in good agreement with the exact result of Rayleigh, as required. Further, the growth of waves with length in a certain range is shown to be suppressed by making this approximate allowance for finite thickness. However, it is found that very short waves, which are quite outside the range of validity of the equation but which are likely to be excited in a numerical integration of the equation, are spuriously amplified as in the case of Moore's equation. Thus, numerical integration of the equation will require use of smoothing techniques to suppress this spurious growth of short wave disturbances.     

Absence of positive solutions of a second-order semilinear parabolic equation with time-periodic coefficients

We consider a parabolic equation with time-periodic measurable bounded coefficients and with a nonlinear source. The equation is studied on the time line and in a domain of x that does not contain the closed unit ball. The definition of a weak solution of the equation is given.     

Long Internal Waves in Fluids of Great Depth

An equation is derived that governs the evolution in two spatial dimensions of long internal waves in fluids of great depth. The equation is a natural generalization of Benjamin's (1967) one-dimensional equation, and relates to it in the same way that the equation of Kadomtsev and Petviashvili relates to the Kortewegde-Vries equation. The stability of one-dimensional solitons with respect to long transverse disturbances is studied in the context of this equation. Solitons are found to be unstable with respect to such perturbations in any system in which the phase speed is a minimum (rather than a maximum) for the longest waves. Internal waves do not have this property, and are not unstable with respect to such perturbations.     

实际气体的幂律状态方程

为克服Onnes(昂内斯)气体状态方程参数多的缺点, 提出了描述实际气体的幂律状态方程.该模型仅包含两个参数,其中幂律函数的阶数可以为任意实数,刻画了实际气体偏离理想气体的程度.满足幂律状态方程的实际气体称为幂律气体.应用于描述氮气（N₂）和四氟甲烷（CF₄）两种实际气体的研究表明,与Onnes气体状态方程相比,幂律气体状态方程可以用较少的参数,准确地描述气体状态方程中压强和体积的幂律关系.此外,温度越低,幂律函数的阶数越小,反映了气体的实际状态越偏离理想气体.     

A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation

A Taylor matrix method is proposed for the numerical solution of the two-space-dimensional linear hyperbolic equation. This method transforms the equation into a matrix equation and the unknown of this equation is a Taylor coefficients matrix. Solutions are easily acquired by using this matrix equation, which corresponds to a system of linear algebraic equations. As a result, the finite Taylor series approach with three variables is obtained. The accuracy of the proposed method is demonstrated with one example.     

果树干周生长模型的研究

生长模型是定量研究果树干周生长过程的有效手段.果树干周生长规律常用L og istic方程、M itscherlich方程、三参数和四参数的R ichards方程、三角函数方程来模拟.本文首次用拟W e ibu ll方程模拟果树干周的生长规律,结果表明,W e ibu ll方程达到了与果树干周最佳拟合的效果,同时,本文方法可先给定由实际情况(如立地条件等)所确定的干周生长的最大值,这使得在研究中可以更好地预测果树将来的干周增长.另外,考虑到因实际情况所造成的初始干周不同这种普遍现象及其引起的生长过程的差异,本文直接引入与单株树体有关的基准干周作为参数,建立了与实际生长情况相结合的干周生长模型,获得了满意的结果.     

由退化抛物方程支配串联系统的零能控性

该文讨论了由一个半线性退化抛物方程与半线性热方程构成的串联系统的零能控性. 这里控制函数仅施加在一个方程上. 证明的关键是建立适当的能观性不等式.     

Solution of the cauchy problem for a hyperbolic equation with constant coefficients in the case of two independent variables

On the plane, we consider a linear partial differential equation of arbitrary order of hyperbolic type. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. For the equation, we obtain an analytic form of the general solution, from which we single out the unique classical solution of the Cauchy problem.     

BOUNDARY LAYER ESTIMATION OF A SINGULAR PROBLEM WITH LIMIT EQUATION OF ORDER 2

ＢＯＵＮＤＡＲＹＬＡＹＥＲＥＳＴＩＭＡＴＩＯＮＯＦＡＳＩＮＧＵＬＡＲＰＲＯＢＬＥＭＷＩＴＨＬＩＭＩＴＥＱＵＡＴＩＯＮＯＦＯＲＤＥＲ２ＨＥＣＨＥＮＧ（何成）ＺＨＡＮＧＷＥＩＴＡＯ（张维弢）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，Ｃｈｉｎｅ...     

Solutions of Riccati-Abel equation in terms of third order trigonometric functions

Solutions of the generalized Riccati equations with third order nonlinearity, named as Riccati-Abel equation, are expressed via third order trigonometric functions. It is shown, as the ordinary Riccati equation, also the Riccati-Abel equation has a relationship with a linear differential equations. A summation formula for solutions of Riccati-Abel equation is established. Possible applications of this formula in the generalized dynamics is outlined. The method admits an extension to the case of generalized Riccati equations with any order of nonlinearity     

On the asymptotics of a solution to an equation with a small parameter at some of the highest derivatives

We study the asymptotic behavior of a solution of the first boundary value problem for a second-order elliptic equation in a nonconvex domain with smooth boundary in the case where a small parameter is a factor at only some of the highest derivatives and the limit equation is an ordinary differential equation. Although the limit equation has the same order as the initial equation, the problem is singulary perturbed. The asymptotic behavior of its solution is studied by the method of matched asymptotic expansions.     

Solutions of nonlinear differential equations

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions. The solution of such an equation is a new generalized function. In this article we formulate necessary and sufficient conditions for when the solution of the given equation in the algebra of new generalized functions is associated with an ordinary function. Moreover, a class of all possible associated functions is described.     

Exact solutions of a two-dimensional nonlinear Schrödinger equation

In the present study, we converted the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved. The general solution of the latter equation in ζ leads to a general solution of NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.     

一类非线性的偏微分方程的解

给出了一类带有时滞的偏微分方程.该方程描述得是含有非局部和时滞边界条件的分布参数系统.运用泛函分析和积分方程的理论,证明了方程解的存在唯一性,得到解的解析表达式.     

Periodic solutions of retarded functional differential equations

The classical criterion for the existence of a periodic solution of a forced, linear, retarded functional differential equation (rfde) references the solution of an adjoint equation which is an advanced functional differential equation. This paper introduces a new criterion which does away with the (anti-causal) advanced functional differential equation.     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. This revised version was published online in June 2006 with corrections to the Cover Date.