A comparative study of ADI splitting methods for parabolic equations in two space dimensions

The main purpose of the paper is a numerical comparison of three integration methods for semi-discrete parabolic partial differential equations in two space variables. Linear as well as nonlinear,equations are considered. The integration methods are the well-known ADI method of Peaceman and Rachford, a global extrapolation scheme of the classical ADI method to order four and a fourth order, four-step ADI splitting method.     

Invariant foliations for parabolic equations

It is proved for parabolic equations that under certain conditions the weak (un-) stable manifolds possess invariant foliations, called strongly (un-) stable foliations. The relevant results on center manifoids are generalized to weak hyperbolic manifolds     

关于一些数值求积公式的渐近性

该文给出了一些数值求积公式的渐近性质,这些公式包括求积分的矩形法则、梯形法则和抛物线法则,包含于余项中的中介点的位置当积分区间的长度趋于零时被确定,对应于该法则的校正公式被得到,它们具有较高的代数精度,我们也进行了一些数值试验,得到较满意的数值结果。     

抛物型域上的周期点

本文证明抛物型域的边界上有且仅有一个有理中性周期点。     

A probabilistic approach to the solution of the Neumann problem for nonlinear parabolic equations

A number of new layer methods for solving the Neumann problemfor semilinear parabolic equations are constructed by usingprobabilistic representations of their solutions. The methodsexploit the ideas of weak-sense numerical integration of stochasticdifferential equations in a bounded domain. In spite of theprobabilistic nature these methods are nevertheless deterministic.Some convergence theorems are proved. Numerical tests on theBurgers equation are presented.     

The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n?3, σ>−2/n and p>max{1,1+σ}, we obtain that pc=n(1+σ)/(n−2) is the critical exponent of these equations. Furthermore, we prove that if max{1,1+σ}<p?pc, then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u₀(x) if p>pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n=1,2, σ>−1 and p>max{1,1+σ}.     

Finiteness of orbit structure for real flag manifolds

Let G be a reductive real Lie group, an involutive automorphism of G, and L=G the fixed point set of . It is shown that G has only finitely many L-conjugacy classes of parabolic subgroups, so if P is a parabolic subgroup of G then there are only finitely many L-orbits on the real flag manifold G/P. This is done by showing that G has only finitely many L-conjugacy classes of -stable Cartan subgroups. These results extend known facts for the case where G is a complex group and L is a real form of G.Research partially supported by NSF Grant GP-16651.     

φ160×116×45离轴抛物面镜的磨制

叙述了三块离轴量为140mm,表面波差≤λ/10离轴抛物面镜的磨制工艺及工艺参数,并对此工艺进行讨论。     

A sufficient condition for the existence of a “dead zone” for solutions of degenerate semilinear parabolic equations and inequalities

We consider the solutions of degenerate parabolic equations and inequalities of the formLu-u _t = |u| ^q sgnu and sgnu(Lu−u _t )−|u| ^q ≥0, 0<q<1, with the elliptic operatorL in divergent or nondivergent form. We establish a dependence of the maximum modulus of the solution on the domain and on the equation (inequality) such that this dependence guarantees the existence of a “dead zone” of the solution. In this case, the character of degeneracy is unessential. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 824–831, December, 1996.     

Pseudotrajectories generated by a discretization of a parabolic equation

A semiimplicit discretization of a parabolic equation is considered. The resulting diffepmorphism is shown to be generically Morse-Smale. Uniform bounds for the dimension of its attractor are given and numerical trajectories—including round-off errors—are shown to approximate the attractor.