Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

New travelling wave solutions to the Boussinesq and the Klein–Gordon equations

In this work, many new travelling wave solutions are established for the Boussinesq and the Klein–Gordon equations. The extended tanh method, the rational hyperbolic functions method, and the rational exponential functions method are used to generate these new solutions. The new solutions are bell-shaped solitons, periodic, and complex solutions. The proposed approaches are also applicable to a large variety of nonlinear evolution equations.     

Analytical solitary wave solutions for the nonlinear analogues of the Boussinesq and sixth-order modified Boussinesq equations

Using tanh function and polynomial function methods, analytical solitary wave solutions have been found for the nonlinear analogues of Boussinesq and sixth-order modified Boussinesq equations where the nonlinearity is in the time-derivative term.     

An application of group theory to matrices and to ordinary differential equations

A result concerned with groups is proved, from which several applications can be derived. We estimate e.g. the number of distinct eigenvalues of the Kronecker product and sum of two given matrices A, B, when A as well as B has distinct eigenvalues. We also discuss the order of the linear ODE whose solutions are the products of solutions of two given linear ODEs, when such ODEs are in certain classes.     

An application of group expansion to the Anderson-Bernoulli model

We establish smoothness of the density of states for 1D lattice Schrödinger operators with potential taking values ${\pm\lambda}$ , for ${\lambda}$ in a class of small algebraic numbers and energy ${E \in\,) -2, 2(}$ suitably restricted away from ${\pm2}$ .     

Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension

A weighted energy estimate with tangential derivatives on the light cone is applied for the Cauchy problem of semilinear wave equations with the null conditions in one space dimension. The well-posedness and lifespan of the solutions are considered based on the vector field method.     

Blowup with vorticity control for a 2D model of the Boussinesq equations

We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a vorticity stretching term and a non-local Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.     

New decay estimates for linear damped wave equations and its application to nonlinear problem

We present new decay estimates of solutions for the mixed problem of the equation v_tt?v_xx+v_t=0, which has the weighted initial data [v₀,v₁]∈(H¹₀(0,∞) ∩L^1,γ(0,∞)) × (L²(0,∞)∩L^1,γ(0,∞)) (for definition of L^1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ?^N for u_tt?Δu+u_t=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.     

On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations

In this paper, we study an LES model for the approximation of large scales of the 3D Boussinesq equations. This model is obtained using the approach first described by Stolz and Adams, based on the Van Cittern approximate deconvolution operators, and applied to the filtered Boussinesq equations. Existence and uniqueness of a regular weak solution are provided. Our main objective is to prove that this solution converges towards a solution of the filtered Boussinesq equations, as the deconvolution parameter goes to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

On the suitable weak solutions to the Boussinesq equations in a bounded domain

In this paper we construct the suitable weak solution for the initial-boundary value problem of the Boussinesq equations and obtain some properties for these solutions. Also in the case of a two dimensional space the uniqueness of weak solution is proved.     

Traveling wave solutions for the combustion model of a shear flow in a cylinder

We establish traveling wave solutions for the combustion model of a shear flow in a cylinder. We study two cases: the infinite Lewis number and an arbitrary Lewis number. For the infinite Lewis number, we establish the existence of traveling wave fronts for both non‐minimal and minimal speeds. For an arbitrary Lewis number, we establish the uniform bounds and exponential decay rates. Copyright © 2015 John Wiley & Sons, Ltd.     

An approximation theory of matrix rank minimization and its application to quadratic equations

Matrix rank minimization problems are gaining plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization problem can be approximated to any level of accuracy via continuous optimization (especially, linear and nonlinear semidefinite programming) problems. One of the main results in this paper shows that if the feasible set of the problem has a minimum rank element with the least Frobenius norm, then any accumulation point of solutions to the approximation problem, as the approximation parameter tends to zero, is a minimum rank solution of the original problem. The tractability under certain conditions and convex relaxation of the approximation problem are also discussed. An immediate application of this theory to the system of quadratic equations is presented in this paper. It turns out that the condition for such a system without a nonzero solution can be characterized by a rank minimization problem, and thus the proposed approximation theory can be used to establish some sufficient conditions for the system to possess only zero solution.     

On the asymptotics of a Bessel-type integral having applications in wave run-up theory

Rapidly oscillating integrals of the form

$$I(r,h) = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {{e^{\frac{i}{h}F(r\cos \phi )}}G(r\cos \phi )d\phi ,} $$

where F(r) is a real-valued function with nonvanishing derivative, arise when constructing asymptotic solutions of problems with nonstandard characteristics such as the Cauchy problem with spatially localized initial data for the wave equation with velocity degenerating on the boundary of the domain; this problem describes the run-up of tsunami waves on a shallow beach in the linear approximation. The computation of the asymptotics of this integral as h → 0 encounters difficulties owing to the fact that the stationary points of the phase function F(r cos ?) become degenerate for r = 0. For this integral, we construct an asymptotics uniform with respect to r in terms of the Bessel functions J ₀(z) and J ₁(z) of the first kind.

     

An application of traveling wave analysis in economic growth model

A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance.     

An oscillating operator related to wave equations in the block spaces

In this article, we study certain oscillating multipliers related to Cauchy problem for the wave equations on the Euclidean space and on the torus. We obtain that, at the end point, these operators are bounded from the L _p spaces to certain block spaces.     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

基于errors-in-variables的预测模型及其应用

预测是统计学实际应用的一个主要方面,多元线性回归预测是一种很好的方法,广泛地应用在各种实际领域,但其局限性及不足也是明显的。本文以一种新的观点认识数据,即认为变量的观测里均含有误差,同时认为不应删除经慎重选择进来的解释变量。为此,本文提出了一种新的多元预测方法———多元线性EIV预测。本文还考虑了新预测模型的一个实例应用,并从相对偏差上与多元回归预测进行了比较,从而揭示了多元线性EIV预测的先进性及较好的预测精度。     

On the existence and uniqueness of classical global solution to a type of Boussinesq equations

The existence and uniqueness of classical global solutions to a type of Boussinesq equations with initial and boundary values are studied in this paper. The existence of such solutions is proved by means of compactness theorem and Schauders fixed point theorem, and its uniqueness by the so called energy method.Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

B-transform and its application to a fish-hyacinth model

A new transform proposed by Oyelami and Ale for impulsive systems is applied to an impulsive fish-hyacinth model. A biological policy regarding the growth of the fish and the hyacinth populations is formulated.