The Eigenvalues of the Angular Spheroidal Wave Equation

Approximate relations are obtained between the eigenvalues λ and the ellipticity parameter c² of the angular spheroidal wave equation. Although based on WKBJ methods and the assumption that λ is large, the relations are useful throughout the complex c²-plane. They are exact at c² = 0, and reproduce the standard asymptotic formulas for λ when c² is large. At intermediate values of c², they provide approximations for the square-root branch points of the multivalued function λ(c²) in the complex c²-plane at which adjacent eigenvalues of the same class become equal in pairs. These branch points lie on an infinite sequence of distorted circular rings. Their exact locations have been computed for the first four rings for angular wavenumbers m = 0,…,4.     

Small Ball Asymptotics for the Stochastic Wave Equation

We examine the small ball asymptotics for the weak solution X of the stochastic wave equation

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1, or ∞ (in the variables , or , respectively). The result, which seems to be true for all values of the parameter τ, according to our numerical experiments, is here proved for small values of the parameter τ.     

Uniformly Controllable Schemes for the Wave Equation on the Unit Square

The paper deals with the numerical approximation of the HUM control of the 2D wave equation. Most of the discrete models obtained with classical finite difference or finite element methods do not produce convergent sequences of discrete controls, as the mesh size h and the time step Δt go to zero. We introduce a family of fully-discrete schemes, nondispersive, stable under the condition \(\Delta t\leq h\slash\sqrt{2}\) and uniformly controllable with respect to h and Δt. These implicit schemes differ from the usual explicit one (obtained with leapfrog time approximation and five point spatial approximations) by the addition of terms proportional to h ² and Δt ². Numerical experiments for nonsmooth initial conditions on the unit square using a conjugate gradient algorithm indicate the excellent performance of the schemes.     

Separation of Variables in a Nonlinear Wave Equation with a Variable Wave Speed

We develop a generalized conditional symmetry approach for the functional separation of variables in a nonlinear wave equation with a nonlinear wave speed. We use it to obtain a number of new (1+1)-dimensional nonlinear wave equations with variable wave speeds admitting a functionally separable solution. As a consequence, we obtain exact solutions of the resulting equations.     

关于RN上的纯量曲率方程

我们证明在RN上纯量方程某种新解的存在性．     

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.     

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

Hierarchic Control for the Wave Equation

This paper deals with the hierarchical control of the wave equation. We use Stackelberg–Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves an exact controllability problem. We consider linear and semilinear equations.     

全空间上带Hardy-Sobolev项的拟线性椭圆方程解的存在性

讨论了全空间上一类带Hardy-Sobolev项的拟线性椭圆问题,利用集中紧原理和适当的实验函数,得到并验证了(PS)_c条件,从而证明了该问题非平凡解的存在性.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

On the Growth of Convergence Radii for the Eigenvalues of the Mathieu Equation

It is proved that the convergence radii ρn of the eigenvalues of the Mathieu equation satisfy lim inf ρn/n² > kk′K² = 2.0418., where the modulus k of the complete elliptic integrals is determined by 2E = K.     

Class of Neumann-Type Problems for the Polyharmonic Equation in a Ball

Computational Mathematics and Mathematical Physics - A set of necessary solvability conditions for the class $${{\mathcal{N}}_{k}}$$ of Neumann-type problems for the polyharmonic equation with a...     

Spectral Deviations for the Damped Wave Equation

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the “entropy” that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.     

Boundary Controllability for the Quasilinear Wave Equation

We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.     

单位球上全纯函数的高阶Schwarz-Pick估计*

作者给出了单位球B_n ? Cⁿ到凸区域?? C上全纯函数的高阶Schwarz-Pick估计. 通过引入双曲度量,得到了单位圆盘D到凸区域?上全纯函数的系数估计. 应用该系数估计结果,得到单位球B_n到?内的全纯函数的高阶Schwarz-Pick估计.特别地, 当?是单位圆盘或右半平面时, 得到的结果分别与熟知的结果是一致的.     

广义浅水波方程新的行波解

通过利用新的G展开法,并借助Mathematica计算软件,研究了广义浅水波方程的精确解,获得了该方程的含有多个任意参数的新的显式行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

单位球上的经典Gelfand问题的逼近解

获得了单位球上的经典Gelfand问题的一个逼近解的序列，同时给出了相应的收敛速度。这个逼近程序是从零函数开始的，因此该程序是可行和有效的。