Published 6 issues per year
ISSN Print: 2152-5080
ISSN Online: 2152-5099
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DIFFERENTIAL CONSTRAINTS FOR THE PROBABILITY DENSITY FUNCTION OF STOCHASTIC SOLUTIONS TO THE WAVE EQUATION
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.
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Venturi Daniele, Cho Heyrim, Karniadakis George Em, Mori-Zwanzig Approach to Uncertainty Quantification, in Handbook of Uncertainty Quantification, 2017. Crossref
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Tartakovsky Daniel M., Gremaud Pierre A., Method of Distributions for Uncertainty Quantification, in Handbook of Uncertainty Quantification, 2015. Crossref
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Venturi D., Sapsis T. P., Cho H., Karniadakis G. E., A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468, 2139, 2012. Crossref
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Venturi Daniele, Cho Heyrim, Karniadakis George Em, Mori-Zwanzig Approach to Uncertainty Quantification, in Handbook of Uncertainty Quantification, 2015. Crossref