Randomly perturbed vibrations

Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

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GROWTH RATES OF TOTAL VARIATIONS OF SNAPSHOTS OF THE 1D LINEAR WAVE EQUATION WITH COMPOSITE NONLINEAR BOUNDARY REFLECTION RELATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007138 ◽

2003 ◽

Vol 13 (05) ◽

pp. 1183-1195 ◽

Cited By ~ 35

Author(s):

YU HUANG

Keyword(s):

Wave Equation ◽

Nonlinear Boundary ◽

Chaotic Behavior ◽

Initial Conditions ◽

Distributed Parameter ◽

Linear Wave ◽

Linear Wave Equation ◽

Energy Pumping ◽

Long Time

The linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end is a useful model for studying chaotic behavior in distributed parameter system. In this paper, we study the dynamics of the Riemann invariants (u, v) of the wave equation by means of the total variations of the snapshots on the spatial interval. Our main contributions here are the classification of the growth of total variations of the snapshots of u and v in long-time horizon, namely, there are three cases when a certain parameter enters a different regime: the growth (i) remains bounded; (ii) is unbounded (but nonexponential); (iii) is exponential, for a large class of initial conditions with finite total variations. In particular, case (iii) corresponds to the onset of chaos. The results here sharpen those in an earlier work [Chen et al., 2001].

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On the supercritical defocusing NLW outside a ball

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00576-3 ◽

2021 ◽

Vol 11 (4) ◽

Author(s):

Piero D’Ancona

Keyword(s):

Wave Equation ◽

Large Time ◽

Dirichlet Boundary ◽

Strichartz Estimate ◽

Energy Inequality ◽

Dirichlet Boundary Conditions ◽

Linear Wave ◽

Linear Wave Equation ◽

Penrose Transform ◽

Dependent Potential

AbstractWe study a defocusing semilinear wave equation, with a power nonlinearity $$|u|^{p-1}u$$ | u | p - 1 u , defined outside the unit ball of $$\mathbb {R}^{n}$$ R n , $$n\ge 3$$ n ≥ 3 , with Dirichlet boundary conditions. We prove that if $$p>n+3$$ p > n + 3 and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a Strichartz estimate for the exterior linear wave equation perturbed with a large, time dependent potential.

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Estimates for the controls of the wave equation with a potential

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2017009 ◽

2018 ◽

Vol 24 (1) ◽

pp. 289-309 ◽

Cited By ~ 3

Author(s):

Sorin Micu ◽

Laurenţiu Emanuel Temereancă

Keyword(s):

Wave Equation ◽

Initial Data ◽

Space Variable ◽

Linear Wave ◽

Linear Wave Equation ◽

One Dimensional ◽

Boundary Controls ◽

The One ◽

Variable Potential ◽

Uniformly Bounded

This article studies the L2-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential a = a(x). It is known these controls depend on a and their norms may increase exponentially with ||a||L∞. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal L2−norm controls are uniformly bounded with respect to the potential a, if the initial data have only sufficiently high eigenmodes.

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A Stability Property of Stochastic Vibration

Journal of Applied Probability ◽

10.1017/s0021900200003089 ◽

2007 ◽

Vol 44 (02) ◽

pp. 444-457

Author(s):

M. Elshamy

Keyword(s):

Wave Equation ◽

Random Fields ◽

Stability Property ◽

Space Variable ◽

Time Variable ◽

Initial Condition ◽

Brownian Sheet ◽

Random Forcing ◽

Continuity Properties ◽

Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

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A Stability Property of Stochastic Vibration

Journal of Applied Probability ◽

10.1017/s0021900200117942 ◽

2007 ◽

Vol 44 (02) ◽

pp. 444-457

Author(s):

M. Elshamy

Keyword(s):

Wave Equation ◽

Random Fields ◽

Stability Property ◽

Space Variable ◽

Time Variable ◽

Initial Condition ◽

Brownian Sheet ◽

Random Forcing ◽

Continuity Properties ◽

Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

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Stochastic models of damped vibrations

Journal of Applied Probability ◽

10.2307/3214993 ◽

1996 ◽

Vol 33 (4) ◽

pp. 1159-1168 ◽

Cited By ~ 7

Author(s):

Maged Elshamy

Keyword(s):

Random Fields ◽

Space Variable ◽

Displacement Function ◽

Time Variable ◽

Stochastic Disturbances ◽

Large Deviations Principle ◽

Stochastic Perturbations ◽

Random Forcing ◽

Non Gaussian ◽

Random Fluctuations

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

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Stochastic models of damped vibrations

Journal of Applied Probability ◽

10.1017/s0021900200100555 ◽

1996 ◽

Vol 33 (04) ◽

pp. 1159-1168 ◽

Cited By ~ 2

Author(s):

Maged Elshamy

Keyword(s):

Random Fields ◽

Space Variable ◽

Displacement Function ◽

Time Variable ◽

Stochastic Disturbances ◽

Large Deviations Principle ◽

Stochastic Perturbations ◽

Random Forcing ◽

Non Gaussian ◽

Random Fluctuations

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

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A Solution for the Interior Sound Field With a Nonuniform Temperature Distribution in the Vehicle Passenger Room

10.1115/imece2001/de-23246 ◽

2001 ◽

Author(s):

Jiuhui Wu ◽

Hualing Chen

Keyword(s):

Temperature Field ◽

Wave Equation ◽

Temperature Distribution ◽

Theoretical Foundation ◽

Initial Conditions ◽

Sound Field ◽

Linear Wave ◽

Nonuniform Temperature ◽

Linear Wave Equation ◽

Nonuniform Temperature Distribution

Abstract A non-linear wave equation in a nonuniform sound space is deduced to determine the coupling relation between sound field and temperature field. Adopting Poisson’s integral a formula for calculating the interior sound field with uneven temperature field and certain initial conditions is developed. This establishes the corresponding theoretical foundation for studying the heat and vibroacoustic comfortability in a vehicle passenger room.

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A Stability Property of Stochastic Vibration

Journal of Applied Probability ◽

10.1239/jap/1183667413 ◽

2007 ◽

Vol 44 (2) ◽

pp. 444-457

Author(s):

M. Elshamy

Keyword(s):

Wave Equation ◽

Random Fields ◽

Stability Property ◽

Space Variable ◽

Time Variable ◽

Initial Condition ◽

Brownian Sheet ◽

Random Forcing ◽

Continuity Properties ◽

Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

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