The weak damped oscillation in a random medium

The paper gives a solution of the random differential equation with random coefficient, that is , where K (t) is a random process and is a damp coefficient, it is a little parameter.     

The Experimental Approach to Effective Stress Law of Coal Mass by Effect of Methane

Effective stress law of all kinds of coal samples, including steam coal, fat coal, corking coal, thin coal and anthracite, under pore pressure of gas, is experimentally studied using a newly developed test machine. These samples are taken from Coal Mines in Wuda, Hebi, Yanzhou, Yangquan, Qingshui, and Gujiao in China. The experiment results show that, under pore pressure of gas, the tested coal samples comply with Biots effective stress law, where the Biots coefficient is not a constant, and is bilinear function of volumetric stress () and pore pressure (p), that is, We define four areas according to the numerical feature of , that is, functionless area of pore pressure, normal function area, fracturing function area, and quasi-soil function area. The effective stress law of coal mass introduced by this paper is a constitutive equation in the study of coupled solid and fluid. This has significance in the drainage and outburst of methane in coal seam.     

Analytical solutions of one-dimensional discrete dynamical systems with chaotic behaviour

This paper studies exact analytical solutions in closed form of the difference equation

Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor (x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector (x) in terms of the gradient of the mean hydraulic head (x), an integrodifferential equation for (x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined.     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

Geometric Analysis of One Singular Cauchy Problem

We consider the Cauchy problem , x(0) = 0, where a ₀₀₀ = 0, a ₀₀₁ = 0, and a ₀₀₂ = 0, and prove the existence of continuously differentiable solutions x(0,] with required asymptotic properties.     

Solvability of Degenerate Semilinear Parabolic Functional Differential Equations

We obtain local and global theorems on the existence and uniqueness of a solution of the semilinear functional differential equation in a Banach space with a parabolic pencil of operators A + B, where the operator A can be noninvertible. Abstract results are applied to partial functional differential equations.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＬｅｍｍａｓＴｈｅｒｅａｒｅｍａｎｙｒｅｓｕｌｔｓａｂｏｕｔｅｘｉｓｔｅｎｃｅ (ｇｌｏｂａｌｏｒｌｏｃａｌ)ａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｏｆｓｏｌｕｔｉｏｎｓｆｏｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ[1- 9].Ｂｙｔｈｅａｉｄｓｏｆｒｅｓｕｌｔｓ[2 ,3]ｏｆｅｑｕａｔｉｏｎ ｕ/ ｔ=Δｕ-λ｜ｕ｜γ- 1ｕｗｉｔｈｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｓ,ｐａｐｅｒ [4 ]ｓｔｕｄｉｅｄｔｈｅｐｒｏｂｌｅｍｏｆ ｕ/ ｔ=Δｕ-λ｜ｅβｔｕ｜γ- …     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

Floquet bundles for scalar parabolic equations

For linear scalar parabolic equations such as on a finite interval 0x, with various boundary conditions, we obtain canonical Floquet solutions u _n (t, x). These solutions are characterized by the property that z(u _n (t, x))=n for all t, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u _n (t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N1 an exponential dichotomy between the bundles span {u _n (·,·)} _n ^=1/N and .     

On the dynamic behaviour of an elastically supported beam of infinite length, loaded by a concentrated force

Summary An elastically supported beam of infinite length, initially at rest, carries a variable concentrated force at a prescribed point A. General expressions are given for the deflection and the bending moment at A (6.3 and 6.4). Three special cases are considered; the first one is defined by =0 for and =K=const. for ; the second one by =0 for 0 > > , given function of for 0 ; the third one applies to problems in which, during the period of impact, itself is an unknown. The results given here may be of use in those railway-engineering problems in which a rail can be considered as a beam of infinite length, and in which the supporting ground has the required properties.     

The linearization of certain class of nonlinear simultaneous equation set containing amplitude and phase frequency characteristics and application

A class of complex function of rational fraction type is frequently used to describe the dynamical properties of systems. It is however quite difficult to establish a mathematical model of this type on the basis of amplitude and phase frequency data collected from experiments conducted on the related physical system. Since the erection of mathematical model G(j) would involve the solution of a set of nonlinear simultaneous equations with the unknown coefficients a_is and b_is(i=0, 1, ..., m, ..., n) in G(j). Up to now, these nonlinear equations have been considered to be very difficult to solve directly. In spite of the fact there are special computer programmes in certain software packages available to tackle this problem, it is by no means an easy task due to the complex procedures involved in picking up a set of initial values that should be close enough to the exact solutions. This paper proposes a simplified method of linearizing these nonlinear equations set so that direct solution is possible. The method can also be applied to systems with factors of (j) andej0 in G(j). An illustration by a workable example is furnished at the end of this paper to show its versatility.     

On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space

We prove the following statement: Theorem 1. Let E and be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every and > 0, there exists a continuous mapping such that

Existence of Positive Solutions for the Singular Equation (ϕ p (y′))′ + g(t,y, y′) = 0

Two theorems concerning the existence of positive solutions for the singular equation are presented. The results are obtained by using the nonlinear Leray-Schauder alternative and the lower-upper solution method.     

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.     

On the Asymptotic Properties of Solutions of Linear Functional Differential Equations with Constant Coefficients and Linearly Transformed Argument

We establish new properties of C ¹(0, +)-solutions of the linear functional differential equation in the neighborhood of the singular point t = +.     

Oscillations and global attractivity in models of hematopoiesis

LetP(t) denote the density of mature cells in blood circulation. Mackey and Glass (1977) have proposed the following equations:

Almost Periodicity in an Ecological Model with <Emphasis Type="Italic">M</Emphasis>-Predators and <Emphasis Type="Italic">N</Emphasis>-Preys by “Pure-Delay Typ” System

By using comparison theorem and constructing suitable Lyapunov functional, we study the following periodic Lotka–Volterra model with M-predators and N-preys by pure-delay type

About a condition for blow up of solutions of cauchy problem for a wave equation

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Periodic Trajectories near Degenerate Equilibria in the Hénon-Heiles and Yang-Mills Hamiltonian Systems

This paper deals with connected branches of nonstationary periodic trajectories of Hamilton equations