High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

Solving inverse initial-value, boundary-value problems via genetic algorithm

There is a growing interest in inverse initial-value, boundary-value (inverse IVBV) problems, and in the development of robust, computationally efficient methods suitable for their solution. Inverse problems are prominent in science and engineering where often an effect is measured and the cause is not known; scientists and engineers observe the response of a system and desire to know the particulars of the system that elicited such a response. IVBV problems result when the equations that govern the behavior of a system are partial differential equations (wave phenomena, diffusion, potential of all kinds, etc.). Thus, inverse IVBV problems stem from systems governed by partial differential equations in which a response has been measured and a characteristic of the system must be computed. In this paper, an approach to solving inverse IVBV problems is presented in which the stated problem is transformed into a nonlinear optimization problem which is then solved using a genetic algorithm. Results are presented demonstrating the effectiveness of this approach for solving inverse problems that result from systems governed by three specific partial differential (1) the heat equation, (2) the wave equation, and (3) Poisson’s equation.     

Detection of pollution terms in nonlinear second order wave systems

ABSTRACT

In this article we are concerned by the problem of detection of noisy terms which arise in image processing as for remote sensing problems, and which are modelled by a nonlinear wave hyperbolic equation. Some data are missing, then the sentinel method of J.-L. Lions (1992) is used; it is a particular least square-like method which permits to distinguish between the missing terms and the pollution terms. In particular, the sentinel theory is given here in its more general and more realistic setting for hyperbolic problems: in this case, the observation and the control have their support in different open sets. The problem of finding a sentinel is equivalent to a non classical null-controllability problem for the wave equation that we solve.     

Compacton-like and kink-like waves for a higher-order wave equation of Korteweg–de Vries type

In this paper, the bifurcation method of the dynamical and numerical approach to differential equations to study higher order wave equations of Korteweg–de Vries (KdV) type is used. With this methodology we obtain the compacton-like and kink-like wave solutions of the high order KdV type equation. Their implicit expressions are given and their planar graphs are simulated. The results show that the numerical integrations are identical with the theoretical derivations.     

Imaging of layered media in inverse scattering problems for an acoustic wave equation

Two-dimensional (2D) inverse scattering problems for the acoustic wave equation consisting of obtaining the density and acoustic impedance of the medium are considered. A necessary and sufficient condition for the unique solvability of these problems in the form of the law of energy conservation has been established. It is proved that this condition is that for each pulse oscillation source located on the boundary of a half-plane, the energy flow of the scattered waves is less than the energy flux of waves propagating from the boundary of this half-plane. This shows that for inverse dynamic scattering problems in acoustics and geophysics when the law of energy conservation holds it is possible to determine the elastic density parameters of the medium. The obtained results significantly increase the class of mathematical models currently used in solving multidimensional inverse scattering problems. Some specific aspects of interpreting inverse problems solutions are considered.     

Burgers方程的精确孤立波解的符号计算

借助于符号计算Maple,给出了一种构造非线性波动方程行波解的直接代数方法,该方法的主要特点是充分利用Riccati方程.使用此方法得到Burgers方程的多组精确行波解,其中包括一些新的孤立波解,这种方法也适用于求解其它的非线性波动方程（组）.     

Solitary wave solutions for a generalized KdV–mKdV equation with variable coefficients

In this work, a generalized time-dependent variable coefficients combined KdV–mKdV (Gardner) equation arising in plasma physics and ocean dynamics is studied. By means of three amplitude ansatz that possess modified forms to those proposed by Wazwaz in 2007, we have obtained the bell type solitary waves, kink type solitary waves, and combined type solitary waves solutions for the considered model. Importantly, the results show that there exist combined solitary wave solutions in inhomogeneous KdV-typed systems, after proving their existence in the nonlinear Schrödinger systems. It should be noted that, the characteristics of the obtained solitary wave solutions have been expressed in terms of the time-dependent coefficients. Moreover, we give the formation conditions of the obtained solutions for the considered KdV–mKdV equation with variable coefficients.     

Boundary feedback stabilization of the telegraph equation: Decay rates for vanishing damping term

We study a semilinear mildly damped wave equation that contains the telegraph equation as a special case. We consider Neumann velocity boundary feedback and prove the exponential stability of the closed loop system. We show that for vanishing damping term in the partial differential equation, the decay rate of the system approaches the rate for the system governed by the wave equation without damping term. In particular, this implies that arbitrarily large decay rates can occur if the velocity damping in the partial differential equation is sufficiently small.     

大规模真实感固流交互实时绘制方法

针对大规模水面固流交互模拟过程中物体运动和水波扩散不真实且实时性差的问题,提出一种大规模水环境实时交互绘制方法.首先采用分区模拟的方法取代传统的整体高度细节方法,实时模拟大型水域环境,降低计算复杂度;然后在物体运动过程中引入波浪力,结合自定义风力因子实现物体随波逐流的运动效果;在交互波生成过程中,引入正态分布函数对物体三角形面片进行计算,以增强水面波生成的真实感;最后采用结合流体黏度和表面污染形成的衰减公式来改进波动方程,使水波扩散效果更加平滑稳定.实验结果表明,在保证实时性的基础上,该方法能更真实地模拟物体在水面上的运动,同时有效地表现交互过程中水波的生成与扩散过程.     

A small dispersion limit to the Camassa–Holm equation: A numerical study

In this paper we take up the question of a small dispersion limit for the Camassa–Holm equation. The particular limit we study involves a modification of the Camassa–Holm equation, seen in the recent theoretical developments by Himonas and Misio?ek, as well as the first author, where well-posedness is proved in weak Sobolev spaces. This work led naturally to the question of how solutions actually behave in these modified equations as time evolves. While the dispersive limit studied here is inspired by the work of Lax and Levermore on the zero dispersion limit of the Korteweg–de Vries equation to the inviscid Burgers’ equation, here there is no known Inverse Scattering theory. Consequently, we resort to a sophisticated numerical simulation to study two representative (one smooth and one peakon), but by no means exhaustive, initial conditions in the modified Camassa–Holm equation. In both cases there appears to be a strong limit of the modified Camassa–Holm equation to the Camassa–Holm equation as the dispersive parameter tends to zero, provided that solutions have not evolved for too long (time sufficiently small). For the smooth initial condition considered, this time must be chosen before the solution approaches steepening; beyond this time the computed solution becomes increasingly complicated as the dispersive term tends towards zero, and there does not appear to be a limit. By contrast, for the peakon initial condition this limit does appear to exist for all times considered. While in many cases the computations required few discretization points, there were some very challenging cases (particularly for the small dispersion computations) where an enormous number of unknowns were required to properly resolve the solution.     

Stochastic quasilinear viscoelastic wave equation with degenerate damping and source terms

In this paper, we consider a stochastic quasilinear viscoelastic wave equation with degenerate damping and source term. We prove the blow-up of solution for stochastic quasilinear viscoelastic wave equation with positive probability or explosive in energy sense.     

Distributed feedback design for systems governed by the wave equation

This article deals with the geometric control of a one-dimensional non-autonomous linear wave equation. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then, based on geometric control concepts, a distributed control law that enforces the exponential stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modelled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation by considering both tracking and disturbance rejection problems. The robustness of the controller has also been studied when the string tension is subjected to sudden variations.     

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrödinger system.     

Method of calculating the collision integral and solution of the Boltzmann kinetic equation for simple gases,gas mixtures and gases with rotational degrees of freedom

The conservative method of calculating the Boltzmann collision integral for simple gases, gas mixtures and gases with rotational degrees of freedom of molecules is presented. In all cases the common approach based on the projection technique of summing up the contributions in the collision integral is used. The method is applied for solving the Boltzmann kinetic equation for two fundamental rarefied gas flow problems: the heat transfer problem and the problem of shock wave structure. A comparison with experimental and numerical data of other authors is reported. It is shown that the considered numerical method allows one to solve the Boltzmann equation for real gases with high accuracy.     

具有动态边界阻尼的波方程的降阶型差分半离散化的 一致指数稳定性

由于在最优控制的数值计算和可观性的反问题中,离散系统关于离散化参数的一致指数稳定性具有重要作用,因而一致指数稳定性得到广泛而深入的研究.众所周知,对指数稳定的波方程的空间变量用经典的有限差分或有限元离散化后,离散格式产生高频病态伪特征模,致使离散系统不是一致指数稳定的.为了恢复一致指数稳定性,学者们引入了添加数值粘性项法和滤波法等方法.然而对于具有动态边界条件的波方程的一致指数稳定性问题研究的较少.本文用降阶型差分方法对此问题进行研究,先对具有动态边界条件的波方程进行降阶处理,然后利用有限差分对其进行空间半离散化,不用再对其进行任何处理,引入合适的李雅普诺夫函数即可验证离散系统是一致指数稳定的.     

Energy control of distributed parameter systems via speed-gradient method: case study of string and sine-Gordon benchmark models

Energy control problems are analysed for infinite dimensional systems. Benchmark linear wave equation and nonlinear sine-Gordon equation are chosen for exposition. The relatively simple case of distributed yet uniform over the space control is considered. The speed-gradient method for energy control of Hamiltonian systems proposed by A. Fradkov in 1996, has already successfully been applied to numerous nonlinear and adaptive control problems is presently developed and justified for the above partial differential equations (PDEs). An infinite dimensional version of the Krasovskii–LaSalle principle is validated for the resulting closed-loop systems. By applying this principle, the closed-loop trajectories are shown to either approach the desired energy level set or converge to a system equilibrium. The numerical study of the underlying closed-loop systems reveals reasonably fast transient processes and the feasibility of a desired energy level if initialised with a lower energy level.     

半波振子天线矩量法的研究

矩量法是将连续方程离散为代数方程组的方法,此法对于求解微分方程和积分方程均适用,本文以半波振子天线为例,系统的阐述了半波振子天线的海伦积分方程的建立,利用矩量法求解海伦积分方程而得半波振子天线上的电流分布,已知电流分布求解半波振子天线在远区的电场表达式和方向图。     

Painlevé analysis,soliton solutions and lump-type solutions of the (3+1)-dimensional generalized KP equation

The Painlevé analysis is applied and the multi-soliton criterion is presented to test the integrability of the (3+1)-dimensional generalized KP equation derived from a Hirota bilinear equation. It is shown that the considered equation does not pass the well known Painlevé test and it is only integrable in a conditional sense. Solitary wave solutions are shown to interact each other like solitons in multiple wave collisions unless some additional conditions are imposed. Moreover, we analyze a class of analytical rational lump-type solutions in detail, which are generated from positive quadratic polynomial function and rationally localized in many directions in the space, based upon the Hirota bilinear form.     

Non-polynomial splines method for numerical solutions of the regularized long wave equation

In this paper, we construct a numerical method based on cubic splines in tension for solving regularized long wave equation. The truncation error is analysed and the method shows that by choosing suitably parameters we can obtain various accuracy schemes. Numerical stability of the method has been studied by using a linearized stability analysis. Test problems are dealt with. The numerical simulations can validate and demonstrate the advantages of the method.     

A matrix formulation to the wave equation with non-local boundary condition

Recently, various sequential numerical schemes have been proposed for the solution of non-classical hyperbolic initial value problems which involve non-local integral terms over the spatial domain. In this paper, we focus on the wave equation with the non-local boundary condition. Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution. Several numerical examples and also some comparisons with another methods will be investigated to confirm the efficiency of this procedure.