HIGHER ORDER ASYMPTOTIC SOLUTION OF MODE Ⅰ CRACK GROWTH IN ELASTIC-PERFECTLY PLASTIC MEDIA

A higher order asymptotic solution of near-tip field is studied for plane-atrain Mode-Ⅰ quasi-static steady crack growth in the incompressible (v=1/2) elastic perfectly-plastic media.The results show that the near-tip stress and strain are fully continuous,and the strain possesses In (A/r) singularity at the crack tip.The expressions of the stress,strain and velocity in each region are also given.

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

In this paper,we study the existence,uniqueness and asymptotic stabgility of the periodic solution for a class of the most,universal fourth-order nonlinear nonautonomous periodic systems.We give the relevant Liapunov function by using the method of analogical slowly changing coefficients.We also give a considerably accurate estimation of the slowly changing coefficients and obtain the sufficient conditions which guarantee the existence,uniqueness and asymptotic Stability of the periodci solutions.

Asymptotic Theory for a General Second-Order Differential Equation

An asymptotic theory developed for a second-order differential equation. We obtain the form of solutions for some class of the coefficients for large x.

Optimal regularity of positive solutions of the Hénon-Hardy equation and related equations

We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈R.It is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω{0}) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.

REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.

Periodic Solutions of a Cooperative System with State Feedback Control

In this paper, we propose a semi-continuous dynamical system to study the cooperative system with feedback control. Based on geometrical analysis and the analogue of Poincare criterion, the existence and stability of the positive order one periodic solutions are given. Numerical results are carried out to illustrate the feasibility of our main results.

Non-Regular Example of Confidence-Interval Construction

When dealing with a regular (fixed-support) one-parameter distribution, the corresponding maximum-likelihood estimator (MLE) is, to a good approximation, normally distributed. But, when the support boundaries are functions of the parameter, finding good approximation for the sampling distribution of MLE (needed to construct an accurate confidence interval for the parameter’s true value) may get very challenging. We demonstrate the nature of this problem, and show how to deal with it, by a detailed study of a specific situation. We also indicate several possible ways to bypass MLE by proposing alternate estimators;these, having relatively simple sampling distributions, then make constructing a confidence interval rather routine. .

Non-Regular Example of Confidence-Interval Construction

When dealing with a regular (fixed-support) one-parameter distribution, the corresponding maximum-likelihood estimator (MLE) is, to a good approximation, normally distributed. But, when the support boundaries are functions of the parameter, finding good approximation for the sampling distribution of MLE (needed to construct an accurate confidence interval for the parameter’s true value) may get very challenging. We demonstrate the nature of this problem, and show how to deal with it, by a detailed study of a specific situation. We also indicate several possible ways to bypass MLE by proposing alternate estimators;these, having relatively simple sampling distributions, then make constructing a confidence interval rather routine. .

Multiplicity of Periodic Solutions for Second Order Hamiltonian Systems with Asymptotically Quadratic Conditions

A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper.Using Fountain Theorem,one multiplicity result of periodic solutions is obtained,which improves some previous results.

THIRD-ORDER NONLINEAR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM

Third order singulary perturbed boundary value problem by means of differential inequality theories is studied. Based on the given results of second order nonlinear boundary value problem, the upper and lower solutions method of third order nonlinear boundary value problems by making use of Volterra type integral operator was established. Specific upper and lower solutions were constructed, and existence and asymptotic estimates of solutions under suitable conditions were obtained. The result shows that it seems to be new to apply these techniques to solving these kinds of third order singularly perturbed boundary value problem. An example is given to demonstrate the applications.

一类位置不变的极值指数估计量

本文提出了一类位置不变的重尾极值估计量,γ∧n(k_(0),k,r)=1/k_(0)∑_(i=1)^(k_(0))(1/r((X_(n-i,n)-X_(n-k,n)/X_(n-k0,n)-X_(n-k,n))^(r)-1))/1+r/k_(0)∑_(i=1)^(k_(0))(1/r(((X_(n-i,n)-X_(n-k,n))/(X_(n-k 0,n)-X_(n-k,n))^(r)-1))其中:γ>0,k 0是小于k的正整数.得到了此位置不变极值估计量的弱相合性和渐近正态性,并根据其渐近展开式得到k 0的最优选择.

基于阿基米德Copula的极尾相依Copula的渐近展开

基于Juri等提出的一阶正规变换条件下极尾相依Copula及其收敛定理,讨论在二阶正规变换的条件下,基于阿基米德Copula的极尾相依Copula的渐近展开.

含临界指标的周期渐近的分数阶Schr dinger-Poisson系统正解的存在性

主要考察含临界非局部项的周期渐近的分数阶Schr dinger-Poisson系统解的存在性,通过运用山路定理和集中紧性原理,最终得到了正解的存在性.

A New Regularizing Algorithm for Solving the First Kind of Fredholm Integral Equations

Singular value system is applied to construct a new class of improved regularizing methods for solving the first kind of Fredholm integral equations with noisy data. By a priori choosing regularizing parameters, optimal convergence order of the regularized solution is obtained. And with aids of MATLAB software, numerical results are presented which roughly coincide with the theoretical analysis.

Regularity and Radial Symmetry of Positive Solutions for a Higher Order Elliptic System

We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi （i = 1, 2） satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1 ＋ p2= q1 ＋ q2 = n＋α/n-α, we classify the solutions of the PDEs system.

EFFECT OF FRACTIONAL ORDER PARAMETER ON THERMOELASTIC BEHAVIORS OF ELASTIC MEDIUM WITH VARIABLE PROPERTIES

This paper is concerned with the thermoelastic behaviors of an elastic medium with variable thermal material properties. The problem is in the context of fractional order heat conduction. The governing equations with variable thermal properties were established by means of the fractional order calculus. The problem of a half-space formed of an elastic medium with variable thermal material properties was solved, and asymptotic solutions induced by a sudden temperature rise on the boundary were obtained by applying an asymptotic approach. The propagations of thermoelastic wave and thermal wave, as well as the distributions of displacement, temperature and stresses were obtained and plotted. Variations in the distributions with different values of fractional order parameter were discussed. The results were compared with those obtained from the case of constant material properties to evaluate the effects of variable material properties on thermoelastic behaviors.

EFFECT OF FRACTIONAL ORDER PARAMETER ON THERMOELASTIC BEHAVIORS IN INFINITE ELASTIC MEDIUM WITH A CYLINDRICAL CAVITY

The thermal shock problems involved with fractional order generalized theory is studied by an analytical method. The asymptotic solutions for thermal responses induced by transient thermal shock are derived by means of the limit theorem of Laplace transform. An infinite solid with a cylindrical cavity subjected to a thermal shock at its inner boundary is studied. The propagation of thermal wave and thermal elastic wave, as well as the distributions of displacement, temperature and stresses are obtained from these asymptotic solutions. The investigation on the effect of fractional order parameter on the propagation of two waves is also conducted.

Smoothness of the Gradient of Weak Solutions of Degenerate Linear Equations

Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1＇2（Ω） defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu ＋ b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.

Singularity of the Extremal Solution for Supercritical Biharmonic Equations with Power-Type Nonlinearity

Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ*> 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r^(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] .