scholarly journals A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities

2017 ◽  
Vol 23 (2) ◽  
pp. 721-749 ◽  
Author(s):  
Fatiha Alabau-Boussouira ◽  
Zhiqiang Wang ◽  
Lixin Yu
Keyword(s):  
Hyperbolic Systems ◽  
Asymptotic Properties ◽  
Single Equation ◽  
Nonlinear Damping ◽  
Energy Decay ◽  
Wave Type ◽  
One Step ◽  
Linear Damping ◽  
Damped Equation ◽  
Appl Math

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in [F. Alabau, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150; F. Alabau, SIAM J. Control Optim. 41 (2002) 511–541; F. Alabau-Boussouira and M. Léautaud, ESAIM: COCV 18 (2012) 548–582] for the linear damping case, and in [F. Alabau-Boussouira, NoDEA 14 (2007) 643–669] for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim. 51 (2005) 61–105; F. Alabau-Boussouira, J. Differ. Equ. 248 (2010) 1473–1517].

scholarly journals Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Akil ◽  
Ibtissam Issa ◽  
Ali Wehbe
Keyword(s):  
Hyperbolic Systems ◽  
Energy Decay ◽  
Coupled Systems ◽  
Bernoulli Beam ◽  
Interpolation Inequalities ◽  
Energy Decay Rate ◽  
Different Types ◽  
Damped Equation ◽  
Frequency Domain Approach ◽  
Euler Bernoulli Beam

<p style='text-indent:20px;'>In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.</p>

scholarly journals Convergence of explicitly coupled simulation tools (co-simulations)

2019 ◽  
Vol 27 (1) ◽  
pp. 23-36
Author(s):  
Thilo Moshagen
Keyword(s):  
Data Exchange ◽  
Equation System ◽  
Fixed Time ◽  
Single Equation ◽  
Step Method ◽  
Time Step ◽  
Simulation Tools ◽  
One Step ◽  
Coupling Time ◽  
Balance Correction

Abstract In engineering, it is a common desire to couple existing simulation tools together into one big system by passing information from subsystems as parameters into the subsystems under influence. As executed at fixed time points, this data exchange gives the global method a strong explicit component. Globally, such an explicit co-simulation schemes exchange time step can be seen as a step of an one-step method which is explicit in some solution components. Exploiting this structure, we give a convergence proof for such schemes. As flows of conserved quantities are passed across subsystem boundaries, it is not ensured that system-wide balances are fulfilled: the system is not solved as one single equation system. These balance errors can accumulate and make simulation results inaccurate. Use of higher-order extrapolation in exchanged data can reduce this problem but cannot solve it. The remaining balance error has been handled in past work by recontributing it to the input signal in next coupling time step, a technique labeled balance correction methods. Convergence for that method is proven. Further, the lack of stability for co-simulation schemes with and without balance correction is stated.

scholarly journals Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

2020 ◽  
Vol 4 (1) ◽  
pp. 400-416
Author(s):  
Abdelmajid Ali Dafallah ◽  
◽  
Fadlallah Mustafa Mosa ◽  
Mohamed Y. A. Bakhet ◽  
Eshag Mohamed Ahmed ◽  
...  
Keyword(s):  
Nonlinear Damping ◽  
Random Attractor ◽  
Beam Equation ◽  
Stochastic Dynamic ◽  
Uniqueness Of Solutions ◽  
Stochastic Dynamical System ◽  
Autonomous Case ◽  
Bounded Absorbing Set ◽  
Linear Damping ◽  
Longtime Behavior

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

scholarly journals Rogue waves and downshifting in the presence of damping

2011 ◽  
Vol 11 (2) ◽  
pp. 383-399 ◽  
Author(s):  
A. Islas ◽  
C. M. Schober
Keyword(s):  
Initial Data ◽  
Rogue Waves ◽  
Nonlinear Damping ◽  
Damping Term ◽  
Nls Equation ◽  
Inverse Spectral Theory ◽  
Damping Parameter ◽  
Spectral Splitting ◽  
Linear Damping ◽  
Further Development

Abstract. Recently Gramstad and Trulsen derived a new higher order nonlinear Schrödinger (HONLS) equation which is Hamiltonian (Gramstad and Trulsen, 2011). We investigate the effects of dissipation on the development of rogue waves and downshifting by adding an additonal nonlinear damping term and a uniform linear damping term to this new HONLS equation. We find irreversible downshifting occurs when the nonlinear damping is the dominant damping effect. In particular, when only nonlinear damping is present, permanent downshifting occurs for all values of the nonlinear damping parameter β. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. We examine the generation of rogue waves in the presence of damping for sea states characterized by JONSWAP spectrum. Using the inverse spectral theory of the NLS equation, simulations of the NLS and damped HONLS equations using JONSWAP initial data consistently show that rogue wave events are well predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. We define δcutoff by requiring that 95% of the rogue waves occur for δ < δcutoff. We find that δcutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur. As a result when damping is present the proximity to homoclinic data and instabilities is more crucial for the development of rogue waves.

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