中立型脉冲微分方程正解的存在性

本文利用Banach压缩映射原理，讨论了中立型时滞脉冲微分方程正解的存在性。     

Probability Interpretation of the Integral of Fractional Order

We establish a relation between stable distributions in probability theory and the fractional integral. Moreover, it turns out that the parameter of the stable distribution coincides with the exponent of the fractional integral. It follows from an analysis of the obtained results that equations with the fractional time derivative describe the evolution of some physical system whose time degree of freedom becomes stochastic, i.e., presents a sum of random time intervals subject to a stable probability distribution. We discuss relations between the fractal Cantor set (Cantor strips) and the fractional integral. We show that the possibility to use this relation as an approximation of the fractional integral is rather limited.     

Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

四阶非线性边值问题解的存在性与上下解方法

该文讨论四阶常微分方程边值问题u^(4)(t)=f(t,u,u″), t∈［0,1］,u(0)=u(1)=u″(0)=u″(1)=0解的存在性, 其中f(t,u,v):［0,1］×R×R→R为Carathéodory函数. 在不限制f关于u,v的增长阶, 不假定f关于u,v的单调性的一般情形下, 用上下解方法获得了解的存在性结果,并讨论了单调迭代求解的有效性.     

Sobolev-Hardy不等式与临界双重调和问题

该文讨论一类带有奇异系数的双重调和方程{△^2u-μu/|x|^s=f(x,u),x∈Ω,u=δu/δv=0,x∈δΩ，这里Ω包含R^N是包含0的有界光滑区域，u∈H0^2(Ω），μ∈R是参数，0≤s≤2,△^2=△△表示双重拉普拉斯算子，当f(x,u)=u^p,p=2N/N-4时，上述问题就是一个临界双重调和问题，该文运用Sobolev-Hardy不等式和变分方法，得到它的解的存在性的一些结果。     

对求解非线性方程方法的探索

BaiCL在2001年提出了双曲函数法，并把解设成具体的双曲函数，即sinhw和coshw的线性组合．本文把解设成是由2个函数f(ξ)和g(ξ)组成的线性组合，它们可以有多种取法，从而使方程更直接、更有效．借助数学软件Maple，用改进后的方法和吴文俊消元法，求解Bai C L文中的一个例子，获得了包含Bai文结果在内的更为丰富、精确的行波解．     

线性泛函方程解的振动性的新结果

本文研究高阶泛函方程x(g(t))=P(t)x(t)+Q1(t)x(g2(t))+…+Qk(t)x(gk+1(t))解的振动性,得到了一些新的振动条件.改进和推广了已有结果.     

论恒力作用下质点的相对论运动

在相对论情况下，讨论了受恒力作用的质点的运动规律。     

基于MATLAB的异步电动机线性化控制系统的仿真

从异步电动机在同步旋转坐标系下的状态方程出发，在一定的条件下，对其系数矩阵简化．推出异步电动机的线性化控制模型．并基于该模型得出异步电动机的线性化控制系统．以给定电机为例，对该模型的有效性、响应用MATLAB进行仿真分析．验证的结果说明上述模型具有实用价值．