含源项浅水波方程的稳态解及其计算方法

Consider the shallow fluid flow in a channel with wavy bottom, which is governed by the following equations: ht (hu)x=0, (hu)t (hu^2 gh^2/2)x=-ghB‘(x),(1)     

On the Expected Number of Zeros of Nonlinear Equations

This paper investigates the expected number of complex roots of nonlinear equations. Those equations are assumed to be analytic, and to belong to certain inner product spaces. Those spaces are then endowed with the Gaussian probability distribution. The root count on a given domain is proved to be ‘additive’ with respect to a product operation of functional spaces. This allows one to deduce a general theorem relating the expected number of roots for unmixed and mixed systems. Examples of root counts for equations that are not polynomials, nor exponential sums are given at the end.     

Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect

In the present study, we start by formally deriving the simplified phenomenological models of long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. These new model equations are analogous to the Green–Naghdi equations, the first-order approximations of the KdV-, or BBM type, respectively. We then justify rigorously that in the long-wave limit, unidirectional solutions of a class of KdV- or BBM type are well approximated by the solutions of the Camassa–Holm equation in a rotating setting. The modeling and analysis of those mathematical models then illustrate that the Coriolis forcing in the propagation of shallow-water waves can not be neglected. Indeed, the CH-approximation with the Coriolis effect captures stronger nonlinear effects than the nonlinear dispersive rotational KdV type. Furthermore, we demonstrate nonexistence of the Camassa–Holm-type peaked solution and classify various localized traveling wave solutions to the Camassa–Holm equation with the Coriolis effect depending on the range of the rotation parameter.

     

On the Shallow Water Equations at Low Reynolds Number

The vertial averages of the incompressible Navier-Stokes equaitons are studied from the point of view of numerical analysis:existence of solution and converagence of algorithms. Three formulations are analysed; existence theorems are obtained when the Reynolds number is small. Convergene of a time implict algorithm is shown, while discretization in space is achived with the finite element method Résumé On étudie, due vue de l'analyse numérique, le problé obtenu en Prenant la moyenne verticale des équations de Navier-stokes: on s'intéresse á l'existence de solutions et á la convergence d'algorithmes. Trois formulations sont analysées; on déor$eacute;mes d'existence pour de faibles valeurs du nombrede Reynolds. On prouve la convergence d'un schéma implicite en temps, tandis que la discr$eacute;tisation en espace est effectuée par éléments finis     

Regularized Equations for Numerical Simulation of Flows in the Two-Layer Shallow Water Approximation

Regularized equations describing hydrodynamic flows in the two-layer shallow water approximation are constructed. A conditionally stable finite-difference scheme based on the finitevolume method is proposed for the numerical solution of these equations. The scheme is tested using several well-known one-dimensional benchmark problems, including Riemann problems.     

Exact Step-Like Solutions of One-Dimensional Shallow-Water Equations over a Sloping Bottom

Mathematical Notes -     

关于有限域上一类方程零点个数的估计

By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.     

Common Conserved Densities AKNS Hierarchies of Equations

ＣｏｍｍｏｎＣｏｎｓｅｒｖｅｄＤｅｎｓｉｔｉｅｓＡＫＮＳＨｉｅｒａｒｃｈｉｅｓｏｆＥｑｕａｔｉｏｎｓ￥ＧｕｏＦｕｋｕｉ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｄｏｎｇＭｉｎｉｎｇＩＮｓｔｉｔｕｔｅ）Ａｂｓｔｒａｃｔ：Ａｓｐｅｐｅｒｃ...     

On the Condition Number of Integral Equations in Acoustics using Modified Fundamental Solutions

To overcome the non-uniqueness problem arising in integral equationsfor the exterior boundary-value problems for the Helmholtz equation,Jones recently suggested adding a series of outgoing waves tothe free-space fundamental solution. We present an analysison the appropriate choice of the coefficients occurring in thisseries in order to minimize the condition number of the integralequations.     

具有单面非完整约束的力学系统的Lie对称性与守恒量

研究具有单面非完整约束的力学系统的Ｌｉｅ对称性。给出由Ｌｉｅ对称性得到系统守恒量的条件和守恒量的形式,并研究上述问题的逆问题,即根据系统的已知积分来求相应的Ｌｉｅ对称性,最后举例说明结果的应用。     

On the Modelling of Shallow-Water Waves with the Coriolis Effect

Journal of Nonlinear Science - Consideration herein is a rotation-Camassa–Holm-type equation, which can be derived as an asymptotic model for the propagation of long-crested shallow-water...     

On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations

 This paper is concerned with the problem of counting the number of zeros of bounded nonoscillatory solutions of higher-order linear ordinary differential equations involving a parameter. It is shown that a recent result for the second-order case is still valid for the higher-order case. Received September 20, 2001; in revised form February 8, 2002 Published online July 12, 2002     

Estimate of Condition Number for Some Discrete Ill-Posed Equations

This paper extends the numerical method and the estimate of the condition number for some discrete ill-posed equations with positive definite matrix in [2] to the case with generalized positive definite matrix.     

On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations

 This paper is concerned with the problem of counting the number of zeros of bounded nonoscillatory solutions of higher-order linear ordinary differential equations involving a parameter. It is shown that a recent result for the second-order case is still valid for the higher-order case.     

准坐标下非完整力学系统的Lie对称性和守恒量

研究准坐标下非完整系统的Ｌｉｅ对称性,首先,对准坐标下非完整力学系统定义无限小变换生成元,由微分方程在无限小变换下的不变性,建立Ｌｉｅ对称性的确定方程,得到结构方程并求出守恒量;其次,研究上述问题的逆问题;根据已知积分求相应的Ｌｉｅ对称性,举例说明结果的应用。