Improved third‐order weighted essentially nonoscillatory scheme

A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.     

Cluster growth in particle-conserving cellular automata

A class of simple two-dimensional cellular automata with particle conservation is proposed for easy simulations of interacting particle systems. The automata are defined by the exchange of states of neighboring cells, depending on the configurations around the cells. By attributing an energy to a configuration of cells, we can select significant rules from the huge number of possible rules and classify them into several groups, based on the analogy with a binary alloy. By numerical calculations, cluster growth is found in two kinds of phases which reveal gas-solid coexistence and liquid droplets. Normalized scaling functions are obtained, and dynamical scaling is examined.     

Invariance Analysis and Variational Conservation Laws for the Wave Equation on Some Manifolds

In this paper we discuss symmetries of a nonlinear wave equation that arises as a consequence of some Riemannian metrics of signature −2. The objective of this study is to show how geometry can be responsible in giving rise to a nonlinear inhomogeneous wave equation rather than assuming nonlinearities in the wave equation from physical considerations. We find Lie point symmetries of the corresponding wave equations and give their solutions in two cases. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined.     

Integrable semiclassical deformations of general algebraic curves and associated conservation laws

Based on the Lenard relations, we completely classify integrable deformations of general algebraic curves. We construct the general solution of the Lenard relation from the invariance condition with respect to an element of the Galois group of the curve. We give some examples and also some associated conservation laws. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 458–469, June, 2007.     

Identification of a dissipation coefficient by a variational method

A generalized inverse problem for the identification of the absorption coefficient for a hyperbolic system is considered. The well-posedness of the problem is examined. It is proved that the regular part of the solution is an L ₂ function, which reduces the inverse problem to minimizing the error functional. The gradient of the functional is determined in explicit form from the adjoint problem, and approximate formulas for its calculation are derived. A regularization algorithm for the solution of the inverse problem is considered. Numerical results obtained for various excitation sources are displayed.     

在可交换四元数空间中的双曲方程的特征边值问题

讨论了一阶和二阶双曲复方程的特征边值问题.对其中两种线性方程,分别在不同的情况下获得通解和可解条件.而对于拟线性的二阶双曲复方程,证明了解的存在性和唯一性.     

Studies of Some Curvature Operators in a Neighborhood of an Asymptotically Hyperbolic Einstein Manifold

On an asymptotically hyperbolic Einstein manifold (M,g₀) for which the Yamabe invariant of the conformal structure on the boundary at infinity is nonnegative, we show that the operators of Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the metric g₀. We deduce in the C^∞ case that the image of the Riemann-Christoffel curvature operator is a submanifold in a neighborhood of g₀.     

Lienard系统的同宿轨族与闭轨族

本文讨论Ｌｉｅｎａｒｄ系统解的一些定性性质，得到了存在同宿轨族、闭轨族、双曲扇形和椭圆扇形、正负半轨有界及其与等倾线相交的充要条件或充分条件．     

EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

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Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems

In this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.