求高阶常系数非齐次线性微分方程特解的新方法

求高阶常系数非齐次线性微分方程:y(n)+P1y(n-1)+…+Pny=f(x)(P1,P2,…,Pn是实数)的特解的一种新方法.首先将该方程降为n个一阶非齐次线性微分方程组:其中w1,w2,…,wn是对应的齐次方程的特征方程:tn+P1tn-1+…+Pn=0的n个根.然后得出了求原方程一个特解的迭代公式.     

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form under the very general structural condition . To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P). Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000     

Explicit solutions of the viscous model vorticity equation

Explicit solutions are found for the viscous version of the model vorticity equation recently proposed by P. Constantin, P. D. Lax, and A. Majda: where H(w) is the Hilbert transform of w, and v is a positive constant. Various properties of these solutions, including the fact that they blow up after a finite time, are discussed.     

Differential-geometric structures in the theory of two-dimensional integrable equations

The gauge representations of the integrable generalization of the Heisenberg magnet in (2+1)-dimensional space-time are interpreted in terms of topological charge. Restrictions on the class of solutions to the equation for a two-dimensional magnet are described for which it becomes gauge-equivalent to the Davy-Stuartson equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 75–81, 1991.The author is grateful to P. P. Kulish for stimulating discussions.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,I — Its WKB-theoretic transformation to the Mathieu equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. Our emphasis is put on the analysis of the singularity structure of its Borel transformed WKB solutions near fixed singular points relevant to the two simple poles contained in the potential of the equation. In Part I, we focus our attention on the construction and analytic properties of a WKB-theoretic transformation that transforms an M2P1T equation to an algebraic Mathieu equation. That transformation plays an important role in Part II ([7]) when we discuss the singularity structure of Borel transformed WKB solutions of an M2P1T equation.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,II — Its relevance to the Mathieu equation and the Legendre equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu equation and the Legendre equation.     

Forced liénard equations with nonlocal nonlinearities

We approximate the operator equation Au+g(u)= h by the averaged system Av + PSPJ.v=h ,3 2llvll P Min the Hilbert space &= L (ft) where ft c Rn has finite mass M .We solvethe system explicitly and discuss the dependence of its solutions as g and h are varied.We extend our analysistoan operator equation which contains a damping term     

Construction de solutions ramifiées autour de deux hypersurfaces caractéristiques pour des opérateurs quasi-linéaires

Let P(x,D)u=v be a partial differential equation, where P(x,D) is a quasi-linear operator of second order with simple characteristic hypersurfaces; we get solutions u which are ramified around two simple characteristic hypersurfaces if v is ramified function and D_t1v, D_t2,v are bounded.     

SOME APPLICATIONS OF BOREL TECHNIQUE IN PARTIAL DIFFERENTIAL EQUATIONS

In this work Borel's technique is applied to analyzing the solvability of partial differetialequations.It is proved that if P is analytic-hypoelliptic,f is C~∞,then the problem for Pu=f with any number of conditions on a fixed point is ill-posed.Besides,some other resultson the flat solution to Laplace equation and wave equation are obtained.     

Equivalence and equation solvability problems for the alternating group A4

It is observed in this paper that the complexities of the equivalence and the equation solvability problems are not determined by the clone of the algebra. In particular, we prove that for the alternating group on four elements these problems have complexity in P; if we extend the group by the commutator as an extra operation, then the equivalence problem is coNP-complete and the equation solvability problem is NP-complete.     

Two-level Additive Schwarz Preconditioners for Nonconforming Finite Element Methods

Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap.

     

W-Commensurability

Properties of the set of solutions (P, Q) of the equation [P, Q] = W in a free group are discussed in the paper, here W is any fixed nonempty group word.     

Error estimation of spectral method for solving three-dimensional vorticity equation

Much work has been done for spectral scheme of P.D.E. (see [1]). Recently the author proposed a technique to prove the strict error estimation of spectral scheme for non-linear problems such as K.D.V.-Burgers' equation, two-dimensional vorticity equation and so on ([2]–[4]). In this paper we generalize this technique into three-dimensional vorticity equation. Under some conditions these error estimations imply convergence. The more smooth the solution of P.D.E., the more accurate the approximate solution.The author is     

On connection formulas for asymptotic expansions for some special solutions of the fifth Painlevé equation

Asymptotic expansions for some special solutions of the fifth Painlevé equation for x→0 and for x→+∞ and the corresponding connection formulas are obtained. An example of application of the obtained formulas to the third Painlevé equation is given. Bibliography: 6 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 19–29. Translated by F. V. Andreev.     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell system

In this paper, we study the nonlinear Klein-Gordon equation coupled with the Maxwell equation in the electrostatic case:

(P)     

矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析

考虑非线性矩阵方程X A~*X~(-n)A=P,其中A是m阶非奇异复矩阵,P是m阶Hermite正定矩阵.本文利用不动点理论讨论了该方程Hermite正定解的存在性及包含区间,给出了极大解的性质及求极大,极小解的迭代算法.研究了极大解的扰动问题,利用微分等方法获得了两个新的一阶扰动界,并给出数值例子对所得结果进行了比较说明.     

关于矩阵方程X+A*X-1A=P的解及其扰动分析

考虑非线性矩阵方程X＋A^＊（X^-1）A=P其中A是n阶非奇异复矩阵,P是n阶Hermite正定矩阵．本文给出了Hermite正定解和最大解的存在性以及获得最大解的一阶扰动界,改进了文[5,6]中的部分结论．     

Quantum Heisenberg group and reflection equation. I

The quantum group of upper triangular matrices and the Hopf algebra associated with the reflection equation are constructed for the R-matrix corresponding to the quantum Heisenberg group. Bibliography: 12 titles. Dedicated to P. P. Kulish on the occasion of his 50th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 28–36. Translated by E. V. Damanskinsky.