On Solution Regularity of Linear Hyperbolic Stochastic PDE Using the Method of Characteristics

2012 ◽  
Vol 2 (3) ◽  
pp. 266-276
Author(s):  
Lizao Li
Keyword(s):  
Differential Equations ◽  
Method Of Characteristics ◽  
Random Variable ◽  
Linear Wave ◽  
Stochastic Pde ◽  
Hyperbolic Pde ◽  
Linear Wave Equation ◽  
Solution Regularity ◽  
Generalized Polynomial

AbstractThe generalized Polynomial Chaos (gPC) method is one of the most widely used numerical methods for solving stochastic differential equations. Recently, attempts have been made to extend the the gPC to solve hyperbolic stochastic partial differential equations (SPDE). The convergence rate of the gPC depends on the regularity of the solution. It is shown that the characteristics technique can be used to derive general conditions for regularity of linear hyperbolic PDE, in a detailed case study of a linear wave equation with a random variable coefficient and random initial and boundary data.

Stochastic Kinetics with Wave Nature

2003 ◽  
Vol 17 (17) ◽  
pp. 929-933
Author(s):  
J. S. Hämäläinen ◽  
J. Merikoski
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lorentz Transformation ◽  
Stochastic Models ◽  
Linear Wave ◽  
Partial Differential ◽  
Linear Wave Equation ◽  
Wave Nature ◽  
Stochastic Kinetics

We consider stochastic second-order partial differential equations. We indroduce a noisy non-linear wave equation and discuss its connections, in particular via the Lorentz transformation, with known stochastic models.

Randomly perturbed vibrations

1995 ◽  
Vol 32 (02) ◽  
pp. 417-428 ◽  
Author(s):  
M. Elshamy
Keyword(s):  
Dirichlet Boundary ◽  
Space Variable ◽  
Initial Conditions ◽  
Linear Wave ◽  
Stochastic Perturbations ◽  
Uniform Topology ◽  
Linear Wave Equation ◽  
Random Forcing ◽  
Continuity Properties ◽  
Positive Time

Let u ε(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 u ε(t, x) from u 0(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 u ε(t, x).

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate

Direct measurements of cold and hot plasma composition and EMIC waves in the outer magnetosphere: Implications for inner magnetosphere wave-particle interactions

2021 ◽  
Author(s):  
Justin Lee ◽  
Drew Turner ◽  
Sarah Vines ◽  
Robert Allen ◽  
Sergio Toledo-Redondo
Keyword(s):  
Unstable Wave ◽  
Linear Wave ◽  
Plasma Composition ◽  
Inner Magnetosphere ◽  
Hot Plasma ◽  
Emic Wave ◽  
Direct Measurements ◽  
Wave Numbers ◽  
Emic Waves

<p>Although thorough characterization of magnetospheric ion composition is rare for EMIC wave studies, convective processes that occur more frequently in Earth’s outer magnetosphere have allowed the Magnetospheric Multiscale (MMS) satellites to make direct measurements of the cold and hot plasma composition during EMIC wave activity. We will present an observation and linear wave modeling case study conducted on EMIC waves observed during a perturbed activity period in the outer dusk-side magnetosphere. During the two intervals investigated for the case study, the MMS satellites made direct measurements of cold plasmaspheric plasma in addition to multiple hot ion components at the same time as EMIC wave emissions were observed. Applying the in-situ plasma composition data to wave modeling, we find that wave growth rate is impacted by the complex interactions between the cold as well as the hot ion components and ambient plasma conditions. In addition, we observe that linear wave properties (unstable wave numbers and band structure) can significantly evolve with changes in cold and hot ion composition. Although the modeling showed the presence of dense cold ions can broaden the range of unstable wave numbers, consistent with previous work, the hot heavy ions that were more abundant nearer storm main phase could limit the growth of EMIC waves to smaller wave numbers. In the inner magnetosphere, where higher cold ion density is expected, the ring current heavy ions could also be more intense near storm-time, possibly resulting in conditions that limit the interactions of EMIC waves with trapped radiation belt electrons to multi-MeV energies. Additional investigation when direct measurements of cold and hot plasma composition are available could improve understanding of EMIC waves and their interactions with trapped energetic particles in the inner magnetosphere.</p>

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

scholarly journals Godunov type scheme for the linear wave equation with Coriolis source term

2017 ◽  
Vol 58 ◽  
pp. 1-26 ◽  
Author(s):  
Emmanuel Audusse ◽  
Stéphane Dellacherie ◽  
Minh Hieu Do ◽  
Pascal Omnes ◽  
Yohan Penel
Keyword(s):  
Wave Equation ◽  
Source Term ◽  
Linear Wave ◽  
Linear Wave Equation
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