Volterra积分方程和泛函微分方程的振动性

本文建立了Ｖｏｌｔｅｒａ积分方程和一阶泛函微分方程解的振动准则．     

Non-Autonomous Evolution Equations on Nonsmooth Domair

The solution of a stationary boundary value problem on a domain with conical points has singularities near these points. Here we first consider existence results in appropriate weighted Sobolev spaces in order to incorporate the singularities. We secondly use these results to prove existence, uniqueness and regularity of solutions of non-autonomous second order evolution equations on such domains.     

A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation

A second-order splitting method is applied to a KdV-like Rosenau equation in one space variable. Then an orthogonal cubic spline collocation procedure is employed to approximate the resulting system. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index 1. Error estimates in L² and L^∞ norms have been obtained for the semidiscrete approximations. For the temporal discretization, the time integrator RADAU5 is used for the resulting system. Some numerical experiments have been conducted to validate the theoretical results and to confirm the qualitative behaviors of the Rosenau equation. Finally, orthogonal cubic spline collocation method is directly applied to BBM (Benjamin–Bona–Mahony) and BBMB (Benjamin–Bona–Mahony–Burgers) equations and the well-known decay estimates are demonstrated for the computed solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 695–716, 1998     

Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.     

Trilinear equations, Bell polynomials, and resonant solutions

A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative examples are made by an algorithm using weights of dependent variables.     

A new method of detecting interferogram in differential phase-contrast imaging system based on special structured x-ray scintillator screen

An x-ray scintillator screen with a special structure,functioning as detector and analyser grating,was proposed for collecting the interferogram of differential phase contrast imaging without absorption grating and difficulty of fabrication by a state of the art technique.On the basis of phase grating diffraction,a detecting model of the scintillator screen was built for analysing the phase and absorption information of objects.According to the analysis,a new method of phase retrievals based on two-images and the optimal structure of screen were presented.     

带加法白噪声的随机Boussinesq方程组的解的渐近行为

考虑速度和温度同时在加法白噪声扰动下的随机Boussinesq方程组的解的渐近特征.可以接轨道得到该随机方程组的唯一解,并可以验证该解生成随机动力系统,进而证明了该随机动力系统存在随机吸引子.     

Reflected backward doubly stochastic differential equations with discontinuous coefficients

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) generator. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.     

一维p-Laplace二阶脉冲微分方程的奇异边值问题

脉冲现象作为一种瞬时突变现象,在现代科技各领域的实际问题中是普遍存在的.本文研究具有奇异边值的一维p-Laplace二阶微分方程在脉冲影响下的正解的存在性,介绍了解的一般性存在定理,并用A-A定理和不动点定理证明了一维p-Laplace二阶脉冲微分方程的奇异边值问题的正解存在性定理.     

三层流体系统非线性界面内波传播理论的研究

基于波陡很小的假设, 利用摄动法, 讨论了任意深度的二维不可压缩、无黏性、无旋的三层流体系统. 在刚性上边界、平底不可渗透条件下, 给出了界面内波传播的统一理论以及描述其波剖面的近似非线性演化方程(NEEs). 最后讨论了几种特殊情形下的近似NEEs.结果表明文献导出的理论结果为本文的特殊情形.