Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Uncertainty Quantification
IF: 3.259 5-Year IF: 2.547 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2011003485
pages 195-213

DIFFERENTIAL CONSTRAINTS FOR THE PROBABILITY DENSITY FUNCTION OF STOCHASTIC SOLUTIONS TO THE WAVE EQUATION

Daniele Venturi
Division of Applied Mathematics, Brown University, Providence, Rhode Island 09212, USA
George Em Karniadakis
Division of Applied Mathematics Brown University, 182 George Street Providence, RI 02912, USA

ABSTRACT

By using functional integral methods we determine new types of differential constraints satisfied by the joint probability density function of stochastic solutions to the wave equation subject to uncertain boundary and initial conditions. These differential constraints involve unusual limit partial differential operators and, in general, they can be grouped into two main classes: the first one depends on the specific field equation under consideration (i.e., on the stochastic wave equation), the second class includes a set of intrinsic relations determined by the structure of the joint probability density function of the wave and its derivatives. Preliminary results we have obtained for stochastic dynamical systems and first-order nonlinear stochastic particle differential equations (PDEs) suggest that the set of differential constraints is complete and, therefore, it allows determining uniquely the probability density function of the solution to the stochastic problem. The proposed new approach can be extended to arbitrary nonlinear stochastic PDEs and it could be an effective way to overcome the curse of dimensionality for random boundary and initial conditions. An application of the theory developed is presented and discussed for a simple random wave in one spatial dimension.


Articles with similar content:

Numerical Approach to Parametric Identification of Dynamical Systems
Journal of Automation and Information Sciences, Vol.46, 2014, issue 3
Vagif Maarif oglu Abdullayev, Kamil Rajab ogly Aida-zade
Recovery of State Vector of Nonlinear Dynamic Systems
Journal of Automation and Information Sciences, Vol.39, 2007, issue 9
Vsevolod M. Kuntsevich
On Maximal Initial Data Set in Problems of Practical Stability of Discrete System
Journal of Automation and Information Sciences, Vol.43, 2011, issue 3
Alexander N. Bashnyakov, Ivan V. Khitko , Vladimir V. Pichkur
Application of the "Geometrical-Optical" Asymptotic Method for Accounting the Impacts of a Complex-Shape Boundary of the Random Region on Multidimensional Nonlinear Irregular Thermal Fields
Heat Transfer Research, Vol.36, 2005, issue 8
G. A. Nesenenko
On the Problem of Stability of Impulse Systems of Random Structure with Constant Lag
Journal of Automation and Information Sciences, Vol.43, 2011, issue 9
Victor I. Musurivskiy