scholarly journals Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
N. Smaoui ◽  
B. Chentouf ◽  
A. Alalabi
Keyword(s):  
Nonlinear Boundary ◽  
Burgers Equation ◽  
Stabilization Problem ◽  
Boundary Stabilization ◽  
Wave Approximation ◽  
Long Wave ◽  
Control Schemes ◽  
De Vries ◽  
Long Wave Approximation ◽  
Stability Of The Solution

Abstract In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is $[0,1]$ [ 0 , 1 ] . First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the $L^{2} $ L 2 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

scholarly journals On the Korteweg–de Vries Long-Wave Approximation of the Gross–Pitaevskii Equation II

2009 ◽  
Vol 35 (1) ◽  
pp. 113-164 ◽  
Author(s):  
Fabrice Béthuel ◽  
Philippe Gravejat ◽  
Jean-Claude Saut ◽  
Didier Smets
Keyword(s):  
Pitaevskii Equation ◽  
Wave Approximation ◽  
Long Wave ◽  
De Vries ◽  
Long Wave Approximation

scholarly journals Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg–de Vries–Burgers Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
B. Chentouf ◽  
N. Smaoui ◽  
A. Alalabi
Keyword(s):  
Boundary Control ◽  
Burgers Equation ◽  
Lyapunov Theory ◽  
Physical Parameters ◽  
Control Laws ◽  
Nonlinear Adaptive Control ◽  
Control Schemes ◽  
De Vries ◽  
Adaptive Boundary Control ◽  
Stability Of The Solution

In this paper, we study the nonlinear adaptive boundary control problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is 0,1. Four different nonlinear adaptive control laws are designed for the MGKdVB equation without assuming the nullity of the physical parameters ν, μ, γ1, and γ2 and depending whether these parameters are known or unknown. Then, using Lyapunov theory, the L2-global exponential stability of the solution is proven in each case. Finally, numerical simulations are presented to illustrate the developed control schemes.

Nonlinear dispersion hyperbolic models of continuous media

2007 ◽  
Vol 5 ◽  
pp. 273-278
Author(s):  
V.Yu Liapidevskii
Keyword(s):  
Boussinesq Equations ◽  
Nonlinear Dispersion ◽  
Order Of Accuracy ◽  
Flow Models ◽  
Wave Approximation ◽  
Long Wave ◽  
Primary Model ◽  
Nonequilibrium Flows ◽  
Internal Inertia ◽  
Long Wave Approximation

Nonequilibrium flows of an inhomogeneous liquid in channels and pipes are considered in the long-wave approximation. Nonlinear dispersion hyperbolic flow models are derived allowing taking into account the influence of internal inertia during the relative motion of phases upon the structure of nonlinear wave fronts. The asymptotic derivation of dispersion hyperbolic models is shown on the example of classical Boussinesq equations. It is shown that the hyperbolic approximation of the equations has the same order of accuracy as the primary model.

scholarly journals NUMERICAL MODELS OF HUGE TSUNAMIS OFF THE SANRIKU COAST

1976 ◽  
Vol 1 (15) ◽  
pp. 61
Author(s):  
Toshio Iwasaki
Keyword(s):  
Numerical Models ◽  
Source Model ◽  
Wave Source ◽  
Sanriku Coast ◽  
Wave Approximation ◽  
Long Wave ◽  
Small Wave ◽  
Wave Heights ◽  
Deep Region ◽  
Long Wave Approximation

Although numerical computations of the generation and propagation of tsunamis are successfully achieved in recent years, modeling of their wave sources is still a big problem. Three kinds of, wave source model, that is statistical, oceanographic and fault model, are studied in this paper. It is found that the first model gives reasonable wave heights as shown in the previous paper, the second one presents roughly one half of those for the first model and the last one produces too small wave heights. Based on the analysis of computed results, nature of undulations off from the shore boundary, directivity of wave propagation and the spindle shaped leading part are discussed. Comparing magnitude of various wave parameters for the leading wave along the minor axis of the wave source, it is shown that the long wave approximation modified by the slope effect illustrates the tsunamis in deep region of the sea and the slope effect is most dominant in shallow region.

Nonlinear dynamics of electrostatic Faraday instability in thin films

2018 ◽  
Vol 855 ◽  
Author(s):  
Dipin S. Pillai ◽  
R. Narayanan
Keyword(s):  
Thin Films ◽  
Standing Waves ◽  
Gas Holdup ◽  
Conducting Liquid ◽  
Mode Instability ◽  
Wave Approximation ◽  
Long Wave ◽  
Faraday Instability ◽  
Unstable Configuration ◽  
Long Wave Approximation

The nonlinear evolution of an interface between a perfect conducting liquid and a perfect dielectric gas subject to periodic electrostatic forcing is studied under the long-wave approximation. It is shown that inertial thin films become unstable to finite-wavelength Faraday modes at the onset, prior to the long-wave pillaring instability reported in the lubrication limit. It is further shown that the pillaring-mode instability is subcritical in nature, with the interface approaching either the top or the bottom wall, depending on the liquid–gas holdup. On the other hand, the Faraday modes exhibit subharmonic or harmonic oscillations that nonlinearly saturate to standing waves at low forcing amplitudes. Unlike the pillaring mode, wherein the interface approaches the wall, Faraday modes may exhibit saturated standing waves when the instability is subcritical. At higher forcing amplitudes, the interface may approach either wall, again depending on the liquid–gas holdup. It is also shown that a gravitationally unstable configuration of such thin films, under the long-wave approximation, cannot be stabilized by periodic electrostatic forcing, unlike mechanical Faraday forcing. In this case, it is observed that the interface exhibits oscillatory sliding behaviour, approaching the wall in an ‘earthworm-like’ motion.

Sign in / Sign up

Export Citation Format

Share Document