In this paper hyperbolic partial differential equations with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein–Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equat...

The operability limits of a supersonic combustion engine for an air-breathing hypersonic vehicle are characterized using numerical simulations and an uncertainty quantification methodology. The time-dependent compressible flow equations with heat release are solved in a simplified configuration. Verification, calibration and validation are carried out to assess the ability of the model to repro...

We have considered the problem of using microseismic data to characterize the flow of injected fluid during hydraulic fracturing. We have developed a simple probabilistic physical model that directly ties the fluid pressure in the subsurface during the injection to observations of induced microseismicity. This tractable model includes key physical parameters that affect fluid pressure, rock fai...

1 Prliminaries 2 1.1 Basic Probability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Stochastic Differential Equations (SDE’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Ito integral and basics of Ito calculus . . . . . . . . . . . . . . ....