奇摄动非线性系统初值问题的套层解

本文研究一类二阶非线性系统的初值问题的奇摄动，揭示了其解呈现双重初始层的性质，通过引进不同量级的伸长变量，得到解的一致有效的渐近展开式。  

Symmetric Stable Laws and Stable-Like Jump-Diffusions

Asymptotic expansions are obtained for finite-dimensional symmetricstable distributions. They are used to prove the existence ofcontinuous transition probability densities of stable and stable-likejump-diffusions, and to construct local multiplicative asymptoticsandglobal two-sided estimates for these densities. As a consequence,the distribution of the first passage times for stable jump-diffusionsis estimated and the integral test for the limsup behaviourof their sample paths as t 0 is provided. 1991 MathematicsSubject Classification: 60E07, 60G17, 60J35, 47D07.  

一类奇摄动边值问题解的套层现象

本文揭示了一类三阶半线性方程奇摄动边值问题解的“套层”现象。利用两次伸长变量的迭代，构造了解的形式渐近展开式。再用微分不等式理论，证明了该奇摄动问题渐近解的一致有效性。  

奇摄动四阶椭圆型偏微分方程

本文用微分不等式理论讨论了一类奇摄动四阶椭圆型偏微分方程,得到了在整个区域上一致有效的渐近展开式.  

四阶半线性椭圆型积分微分方程的奇摄动

本文研究了一个四阶椭圆型积分微分方程的奇摄动问题。证明了相应的比较定理并得到了解的一致有效的渐近展开式。  

奇摄动非线性边值问题

本文讨论了一类奇摄动非线性边值问题 .利用伸长变量和边界层校正法 ,得到了问题解的形式渐近展开式 .再用微分不等式理论 ,证明了解的一致有效性  

拟线性常微分方程组边值问题解的估计

本文研究拟线性常微分方程组边值问题x′=f(t,x,y,ε),x(0,ε)=A(ε)εy″=g(t,x,y,ε)y′+h(t,x,y,ε)y(0,ε)=B(ε),y(1,ε)=C(ε)的奇摄动。其中x,f,y,h,A,B和C均属于R~n,g是n×n矩阵函数。在适当的条件下,利用对角化技巧和不动点定理证明解的存在,并估计了余项.  

On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators

In this paper we set out to understand Filon-type quadratureof highly-oscillating integrals of the form ₀¹ f(x) e^ig(x) dx,where g is a real-valued function and >> 1. Employingad hoc analysis, as well as perturbation theory, we demonstratethat for most functions g of interest the moments behave asymptoticallyaccording to a specific model that allows for an optimal choiceof quadrature nodes. Filon-type methods that employ such quadraturenodes exhibit significantly faster decay of the error for highfrequencies . Perhaps counterintuitively, as long as optimalquadrature nodes are used, rapid oscillation leads to significantlymore precise and more affordable quadrature.  

Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported. The first author was supported by China Postdoctoral Sciences Foundation.  

Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.