On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs),   S_mn ( c , η)  , and their eigenvalues,  λ _mn   , for arbitrary complex size parameter c in the asymptotic regime of large  | c |  , m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary  arg( c )  , the PSWFs and their eigenvalues are accurately expressed by established prolate -type or oblate -type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points,   c ^mn _{○; r}   , in the complex c -plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m , n , and  arg( c )  . The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c , are derived for both the prolate - and oblate -type asymptotic expansions and for both  ( n − m )  even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of   c ^mn _{○; r}   . We document this ordering for the specific case of  arg( c ) =π/4  , which occurs for the diffusion equation in spheroidal coordinates. Some representative values of  λ _mn   and   S_mn ( c , η)  for large, complex c are also given.     

Separation of Eigenvalues of the Wave Equation for the Unit Ball in RN

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If W_l denotes the lth positive root for some fixed order x, the minimum gap property assures that {exp(±iw_lt|l = 1, 2,...} form a Riesz basis in L²(0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.     

The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.     

勒让德方程本征值的确定

大部分数学物理方法教材直接给出了勒让德方程本征值的表达式,比较突兀,学生也难以理解.为消除学生在此处的学习障碍,直接从勒让德方程的一般形式入手,通过常点邻域的级数解法,推导出勒让德方程本征值的表示形式.     

Wavelets Based on Prolate Spheroidal Wave Functions

The article is concerned with a particular multiresolution analysis (MRA) composed of Paley–Wiener spaces. Their usual wavelet basis consisting of sinc functions is replaced by one based on prolate spheroidal wave functions (PSWFs) which have much better time localization than the sinc function. The new wavelets preserve the high energy concentration in both the time and frequency domain inherited from PSWFs. Since the size of the energy concentration interval of PSWFs is one of the most important parameters in some applications, we modify the wavelets at different scales to retain a constant energy concentration interval. This requires a slight modification of the dilation relations, but leads to locally positive kernels. Convergence and other related properties, such as Gibbs phenomenon, of the associated approximations are discussed. A computationally friendly sampling technique is exploited to calculate the expansion coefficients. Several numerical examples are provided to illustrate the theory.     

The Wave Equation in Lp-Spaces

In the first part of the paper, Gaussian estimates are used to study $L^p$-summability of the solution of the wave equation in $L^p(\Omega)$ associated with a general operator in divergence form with bounded coefficients. Secondly, we prove that if $\Omega$ is a cube in $\RR^N$, then the Laplacian with Dirichlet or Neumann boundary conditions generates an $\al$-times integrated cosine function on $L^p(\Omega),\;1\le p <\infty$ for any $\al\ge (N-1)|\frac{1}{2}-\frac{1}{p}|$.     

Asymptotic Approximations for Prolate Spheroidal Wave Functions

Uniformly valid (with respect to the independent variable) asymptotic approximations to the radial, prolate spheroidal wave functions are constructed from Bessel-function and Coulomb-wave-function models for large values of the wave number c. The prolate angular functions also are considered, but more briefly. The emphasis is on qualitative accuracy (such as might be useful to the physicist), rather than on efficient algorithms for very accurate numerical computation, and the error factor for most of the approximations is 1 + O (1/c) as c↑∞.     

高阶复微分方程解的超级的角域分布

设f1,f2,…,fn是复方程f(n)+An-1f(n-1)+…+A0f=0的n个线性无关解,其中A0,A1,…,An-1是不全为多项式,且至少有一个为无限级整函数,σ2(Aj)=0(j=1,2,…,n-1).假设E=f1,f2,…,fn.研究了微分方f(n)+An-1f(n-1)+…+A0f=0的解在角域中的零点分布,获得E的超级为+∞的Borel方向与零点聚值线的关系.     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.     

The Stochastic Nonlinear Damped Wave Equation

We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in Rⁿ of the Klein--Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied.     

High-Frequency Asymptotic Expansions for Certain Prolate Spheroidal Wave Functions

Prolate Spheroidal Wave Functions (PSWFs) are a well-studied subject with applications in signal processing, wave propagation, antenna theory, etc. Originally introduced in the context of separation of variables for certain partial differential equations, PSWFs became an important tool for the analysis of band-limited functions after the famous series of articles by Slepian et al. The popularity of PSWFs seems likely to increase in the near future, as band-limited functions become a numerical (as well as an analytical) tool.     

二阶Camassa-Holm方程行波解

对二阶Camassa-Holm方程行波解的情况进行了讨论.利用解的唯一性,得到了如下结论:二阶CH方程的行波解唯一存在,但不具有u(x,t)=kem(x-ct)形式.还为二阶CH方程行波解的研究提供了一种新途径和方法.     

On the Ginzburg-Landau Wave Equation

This note studies the global existence of strong solutions ofthe initial value problem of the Ginzburg-Landau wave equationon Rⁿ (n = 1, 2, 3). Current address: Department of Mathematics and Statistics, Universityof New Mexico, Albuquerque, NM 87131, USA     

p-Adic Analogue of the Wave Equation

In this paper, a p-adic analogue of the wave equation with Lipschitz source is considered. Since it is hard to arrive the solution of the problem, we propose a regularized method to solve the problem from a modified p-adic integral equation. Moreover, we give an iterative scheme for numerical computation of the regularlized solution.