多维区域中非线性偏微分方程的修正Laguerre谱与拟谱方法

研究多维区域中非线性偏微分方程的谱与拟谱方法．建立了修正Laguerre正交逼近与插值结果,这些结果对于建立和分析无界区域中的数值方法起着重要的作用．作为结果的一个应用,研究了二维无界区域中的Logistic方程的修正Laguerre谱格式,证明了它的稳定性和收敛性．数值试验结果表明所提出方法具有很高的精度,与理论分析结果完全吻合．     

Non-Isotropic Jacobi Spectral and Pseudospectral Methods in Three Dimensions

Non-isotropic Jacobi orthogonal approximation and Jacobi-Gauss type interpolation in three dimensions are investigated. The basic approximation results are established, which serve as the mathematical foundation of spectral and pseudospectral methods for singular problems, as well as problems defined on axisymmetric domains and some unbounded domains. The spectral and pseudospectral schemes are provided for two model problems. Their spectral accuracy is proved. Numerical results demonstrate the high efficiency of suggested algorithms.     

MKdV方程的多辛Fourier拟谱方法

基于谱微分矩阵方法,给出MKdV方程的多辛Fourier拟谱格式及其相应多辛离散守恒律,证明了它等价于通常的Fourier拟谱格式．数值结果表明,格式对于长时间计算具有稳定性与高精度．     

用混合Jacobi-球面调和拟谱方法数值求解Fisher型方程

本文提出了一种数值求解单位球内Fisher型方程的全离散混合Jacobi-球面调和拟谱格式,数值结果显示该方法是有效的.     

Holomorphic Spinors and the Dirac Equation

In the first part of this paper, the closed spin Kähler manifolds of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator, are characterized by holomorphic spinors. In the second part, the space of holomorphic spinors on a Kähler–Einstein manifold is described by eigenspinors of the square of the Dirac operator and vanishing theorems for holomorphic spinors are proved.     

傅立叶超函数和扩充傅立叶超函数与爱米特热方程

证明了傅立叶超函数和扩充傅立叶超函数可用爱米特热方程的解来表示,且用以表示的解有很良好的性质.     

Volterra Integral Equation of Hermite Matrix Polynomials

The primary purpose of this paper is to present the Volterra integral equation of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.     

四元矩阵方程AXB＝D的Hermite解

本给出了四元矩阵方程AXB＝D有Hermite解的充要条件，利用A，B，D及它的Moore-penrose逆的一般Hermite解表示。     

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper, a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this approach. We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation, which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.AMS subject classifications: 65M70, 41A30, 81Q05     

守恒方程的Chebychev-Legendre拟谱粘性方法

给出了非线性守恒方程初边值问题的Chebychev-Legendre拟谱粘性法(CLSV). 文中,用补偿方法处理边界条件,而对高频部分使用粘性法,以恢复精度. 最后证明了在适当条件下,CLSV解收敛于唯一的熵解.     

Smooth approximations to solutions of perturbed Weyl and Dirac equations

Iterations of spherical mean values of the initial condition are used to approximate solutions of the Weyl and the Dirac equations in the Hilbert space [L ²(ℝ ⁿ )] ^N . In this work we smooth the potential and the initial condition, without restrictions on the increase rate as ∥x∥ → ∞. By iterating perturbations of such mean values, we prove apriori estimates for the approximate solutions of the perturbed Weyl and Dirac equations in the topology of uniform convergence of time and space derivatives on compact subsets in ℝ×ℝ ⁿ . Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 538–555, October, 1996. I thank Professor A. M. Chebotarev and Professor R. Quezada for their comments and help in the preparation of this paper. This research was partially supported by SNI and CONACyT-E, grant No 0233P-E9506.     

On Two-Dimensional Finite-Gap Potential Schroedinger and Dirac Operators with Singular Spectral Curves

We describe a wide class of two-dimensional potential Schroedinger and Dirac operators which are finite-gap at the zero energy level and whose spectral curves at this level are singular, in particular may have n-multiple points with n3.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Multilevel Iterative Algorithms for Accelerating Spectral Approximations of Integral Equations

In this paper, spectral collocation and Nyström methods for integral equations are discussed. Some multilevel correction schemes are presented for infinitely accelerating the covergence even if the exact solution is not so smooth.     

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.

     

Fundamental Geometric Structures for the Dirac Equation in General Relativity

We present an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures. Structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory. Various aspects can be most neatly understood within this context, and a number of questions can be most properly addressed (specifically in view of the formulation of QFT on a curved background). In particular, we clarify the fact that the usual spinor structure can be weakened while retaining all essential physical aspects of the theory.     

Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

Efficient Laguerre and Hermite Spectral Methods for Odd-Order Differential Equations in Unbounded Domains

Laguerre dual-Petrov-Galerkin spectral methods and Hermite Galerkin spectral methods for solving odd-order differential equations in unbounded domains are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Numerical results demonstrate the effectiveness of the suggested approaches.