Generalized derivatives of a region function and its applications

This paper is based on some fundamental concepts im [7], Clarke’s generalizedderivatives,as well as Lasotra’s and Strauss’s definitions of differential D(x) of amultivalued function f(x).Thereby,the generalized derivatives of a region function F(x) isdefined asD_F(x)=U∩{G(x)(?)B(R), (?)x∈B(R); G(x)=F’_x=F’(x)}The existence of the generalized derivatives of a region function F(x) is discussed:thenecessary and sufficient conditions of existence of the Fréchet generalized derivatives ofsuch a function is established.     

湍流普适平衡区的能谱

本文应用作者之一提出的确定湍流能谱的理论,对八种试探函数和四个波数取样值用方程误差方法数值求解能谱和阻尼系数满足的两个积分方程。结果得到湍流普适平衡区能谱的一个新的表达式。这个新的表达式与作者之一原来得到的表达式形式不同,但二者对应的三维和一维能谱差别很小;这说明了所用理论的自恰性。新的表达式较为简便。     

A class of incompletely exponentially fitted difference schemes for a singular perturbation problem

Some sufficient conditions are considered, under which the solutions of a class of incompletely exponentially fitted difference schemes converge uniformly in e, with orders one and two, to the solution of the singular perturbation problem: eu"+a(x)u’-b(x)u=f(x), for 0a>0, b(x)≥0. From these conditions.an incompletely exponentially fitted second-order scheme is derived. Finally, the results of some numerical experiments are given.     

Nonlinear oscillations and boundary value problems for Hamiltonian systems

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|²+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics.     

Classical solutions of hyperbolic initial boundary-value problems with state-dependent delays

We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order

Traveling Waves of a Singularly Perturbed System of Integral–Differential Equations Arising from Neuronal Networks

A nonlinear nonlocal model arising from synaptically coupled neuronal networks with two integral terms is considered. The existence and stability of several traveling wave solutions are established by using ideas in differential equations and functional analysis. Steady-state solutions of some inhomogeneous integral–differential equations are also investigated. We consider several types of kernel functions: (I) positive functions, such as and , where ρ>0 is a constant; (II) nonnegative kernels with compact supports, for examples, (i) 1$$" align="middle" border="0"> , and (ii) {\pi\over 2}$$" align="middle" border="0"> ; (III) Mexican hat type kernel functions, such as and , where A>B>0 and a>b>0 are constants.Dedicated to Professor Yulin Zhou and Professor Boling Guo on the Occassions of their birthdays.     

高维线性插值方法及其在3维温度场计算中的应用

通过分析1维和2维线性插值可以推导出任意斜角直线坐标系下n维线性插值的一般计算公式以及有唯一解的条件,这一结论能够应用于三维温度场计算。可以将n维插值问题归结如下:已知n 1维空间中的n 1个点的坐标以及第n 2个点的n个坐标分量xn 2,1,x n 2,2,,xn 2,n,求解该点的第n 1个坐标分量xn 2,n 1。根据线性插值定义,第n 2个点位于前n 1个点所确定的n维超平面上。根据这一条件列写方程、求解方程可得到插值xn 2,n 1。n维插值问题有唯一解的条件是已知的n 1个点在n维空间中构成的多面体的体积不为0。推导过程在斜角直线坐标系中完成,因而结论具有较大普适性。     

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＦＯＲＡＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＦＩＲＳＴＯＲＤＥＲＳＹＳＴＥＭＣｈｅｎＳｏｎｇｌｉｎ（陈松林）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ８,１９８４;ＲｅｖｉｓｅｄＡｐｒｉｌ１５,１９...     

THE CRITERION ALGORITHM OF RELATION OF IMPLICATION BETWEEN PERIODIC ORBITS(Ⅱ)

In recent years,there is a wide interest in Sarkovskii’s theorem and related study.According to Sarkovskii’s theorem,if the continuous self-map f of the closed interval has a3-periodic orbit,then f must has an n-periodic orbit for any positive integer n.But f can nothave all n-periodic orbits for some n.Example.LetEvidently f has only one kind of3-periodic orbit in the two kinds of3-periodic orbits,whichexplains that it isn’t far enough to uncover the relation between periodic orbits by theinformation which Sarkovskii’s theorem has offered.In this paper,we raise the concept oftype of periodic orbits,and give a feasible algorithm which decldes the relation ofimplication between the two kinds of periodic orbits.     

Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.     

Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and f, g smooth in all variables.

     

An Elliptic Lindstedt--Poincaré Method for Certain Strongly Non-Linear Oscillators

An elliptic Lindstedt--Poincaré (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form , in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with are studied in detail.     

Relaxation of bulk and interfacial energies

In this paper we obtain an integral representation for the relaxation inBV(Ω; ? ^p ) of the functional $$u \mapsto \int\limits_\Omega {f(x.\nabla u(x))dx + \int\limits_{\sum _{(u)} } {\varphi (x,[u](x),v(x))dH_{N - 1} (x)} }$$ with respect to theBV weak * convergence.     

Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients

The paper concerns the asymptotic behavior as of solutions u(t, x) of the Cauchy problem for the inhomogeneous equation , x>0, with a mass-preserving boundary condition at x=0. It is assumed that F C ¹[0, 1] F(0)=F(1)=0, and that one of the following conditions holds: (i) F has just one zero in (0, 1) and the derivative F is negative at 0 and 1, (ii) F is zero in (0, ) and positive in (, 1) with 0<<1, (iii) F is positive in (0, 1) and c ₀ ² >2F(0) (the pushed case). Here c ₀ is the minimum of possible speeds of travelling waves for the equation with k(x) replaced by zero. Under a natural restriction imposed on k(x) we prove, for a wide class of initial functions, that u(t, x) approaches w(x + m(t)) as uniformly in x>0, where w is a travelling wave with the speed c ₀ and m is a solution of dm/dt = c ₀-k(m). The result has immediate applications to the multidimensional non-linear diffusion equations arising in population genetics, combustion theory, etc.     

On the shape of a slender axisymmetric nonstationary cavity

Very few studies have been made of three-dimensional nonstationary cavitation flows. In [1, 2], differential equations were obtained for the shape of a nonstationary cavity by means of a method of sources and sinks distributed along the axis of thin axisymmetric body and the cavity. In the integro-differential equation obtained in the present paper, allowance is made for a number of additional terms, and this makes it possible to dispense with the requirement ¦ In ¦ 1 adopted in [1, 2]. The obtained equation is valid under the weaker restriction 1. In [3], the problem of determining the cavity shape is reduced to a system of integral equations. Examples of calculation of the cavity shape in accordance with the non-stationary equations of [1–3] are unknown. In [4], an equation is obtained for the shape of a thin axisymmetric nonstationary cavity on the basis of a semiempirical approach. In the present paper, an integro-differential equation for the shape of a thin axisymmetric nonstationary cavity is obtained to order ² ( is a small constant parameter which has the order of the transverse-to-longitudinal dimension ratio of the system consisting of the cavity-forming body, the cavity, and the closing body). A boundary-value problem is formulated and an analytic solution to the corresponding differential equation is obtained in the first approximation (to terms of order ² In ), A number of concrete examples is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 38–47, July–August, 1980.I thank V. P. Karlikov and Yu. L. Yakimov for interesting discussions of the work.     

Stochastic instability of nonlinear oscillations

Consider a dynamic system whose behavior is described by

$$x'' + \omega ^2 _0 (1 + \alpha x^2 )x + F(t)x = 0$$     

Local Minimizers and Quasiconvexity – the Impact of Topology

The aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals : from a purely topological point of view. The basic assumptions on are sequential lower semicontinuity with respect to W^1,p-weak convergence and W^1,p-weak coercivity, and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds and .In the first part of the paper, we focus on the case where is non-contractible and proceed by establishing a link between the latter problem and the question of enumeration of homotopy classes of continuous maps from various skeleta of into . As this in turn can be tackled by the so-called obstruction method, it is evident that our results in this direction are of a cohomological nature.The second part is devoted to the case where =^N and is a bounded smooth domain. In particular we consider integralswhere the above assumptions on can be verified when the integrand F is quasiconvex and pointwise p-coercive with respect to the gradient argument. We introduce and exploit the notion of a topologically non-trivial domain and under this establish the first existence and multiplicity result for strong local minimizers of that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].     

On the boundedness and the stability properties of solution of certain fourth order differential equations

This paper investigates equation(1)in two cases:(i)P≡0,(ii)P(≠O)satisfies|P(t,x,y,z,ω)|≤(A |y| |z| |ω|)q(t),where q(t)is a nonnegative function of t.For case(i)the asymptotic stability in the large of the trivial solution x=0 is investigatedand for case(ii)the boundedness result is obtained for solutions of equation(1).Theseresults improve and include several well-known results.     

A connection formula for the second Painlevé transcendent

We consider the second Painlevé transcendent $$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$ It is known that if y(x) ～ k Ai (x) as x → + ∞, where ?1<k<1 and Ai (x) denotes Airy's function, then $$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$ where the constants d, c depend on k. This paper shows that $$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$ , which confirms a conjecture by Ablowitz & Segur.     

The irrotational waves and the convergence of Levi-Civita's series

Summary The paper shows that the existence of irrotational surface waves established by an investigation of Levi-Civita may be extended till the breaking, but that it is valid when is 0,84 p 1 (and when p₁=np with n whole number). No solution is known till to day when is p<0,84 (and when is 1,00<p₁<<1,68).

---

Sommario La nota estende la dimostrazione dell'esistenza di onde superficiali irrotazionali, data dal Levi Civita per una ampiezza finita ma sufficientemente piccola dimostrando l'esistenza di queste onde fino al frangimento. Tanto questa dimostrazione che quella originaria di Levi Civita sono valide finchè sia 0,84 p 1 (e quando sia p₁=np con n intero). Nessuna soluzione è nota fino ad oggi quando sia p<0,84 (e quando sia 1 p₁ 1,68).