Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10     

High accuracy time and space transform method for advection–diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection–diffusion equation in the unbounded domain Rⁿ is converted to one on the bounded domain [?1, 1]ⁿ, and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright © 2008 John Wiley & Sons, Ltd.     

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana’s (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of N samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to J, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.      

Application of differential transform method to non-linear oscillatory systems

In this paper, the differential transform method is proposed for solving non-linear oscillatory systems. These solutions do not exhibit periodicity, which is the characteristic of oscillatory systems. A modification of the differential transform method, based on the use of Padé approximants, is proposed. We use alternative technique by which the solution obtained by the differential transform method is made periodic. The method is described and illustrated with examples. The results reveal that the method is very effective and convenient.     

Fourier series and integral equation method for the exterior Stokes problem

The exterior Stokes problem between two parallel planes that are separated by a prismatic cylinder is extended to the interior of the prism by requiring the continuity of the velocity across the lateral faces. The well‐posedness of the exterior–interior problem is proved in suitable weighted Sobolev spaces. The solution is represented by Fourier series in the z‐variable. The Fourier coefficients, solutions of auxiliary two‐dimensional exterior–interior problems, are analyzed by viewing them as boundary integral equations of potential theory and global regularity of the densities, is established in weighted Sobolev spaces of traces. A boundary element method, with suitably refined mesh size, is implemented for the numerical treatment of the Fourier coefficients. This provides optimal convergent semi‐ and fully‐discrete spectral methods of Fourier–Galerkin type. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

一端固支一端简支变厚度梁的弹性力学解

研究了一端固支另一端简支连续变厚度梁在静力荷载作用下的应力和位移分布.通过引入单位脉冲函数和Dirae函数,将固支边等效为简支边与未知水平反力的叠加,利用平面应力问题的基本方程,导出满足控制微分方程及左右两端边界条件的位移函数的一般解,对上下表面的边界方程作Fourier级数展开,结合固支边位移为O的条件确定待定系数,得到的解是高精度的.数值结果与商业有限元软件ANSYS进行了比较,显示出很好的精度.     

Parseval Theory of Complex Wavelet Transform for Wavelet Family Including Rotational Parameters

A rotational parameter Rθ has been introduced to complex wavelet transform （CWT）. The rotational CWT （RCWT） corresponds to a matrix element 〈φ｜U2（θ;μ;κ）[F〉 in the context of quantum mechanics, where U2（θ;μ;κ） is a two-mode rotational displacing-squeezing operator in the 〈η｜ representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac＇s representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.     

实现Hough变换的一种光学方法

讨论了Hough变换及其与相关计算的关系 ,提出了基于非相干光相关器的 ,以直线阵列作模板 ,以线光源阵列为照明光源的Houhg变换光学系统。并给出了计算机模拟此系统运算的结果。     

On the solutions of the Moisil–Théodoresco system

A structure theorem is proved for the solutions to the Moisil–Théodoresco system in open subsets of ?³. Furthermore, it is shown that the Cauchy transform maps L₂(?², ?_{0, 2}⁺) isomorphically onto H²(?₊³, ?_{0, 3}⁺), thus proving an elegant generalization to ?² of the classical notion of an analytic signal on the real line. Copyright © 2008 John Wiley & Sons, Ltd.     

关于粗糙奇异积分算子的一点注记

研究R^n上一类沿多项式曲线的奇异积分算子，在一些相当弱的尺寸条件下建立了这些算子的L^p有界性．