Uncertainty characterization for advection/diffusion equations
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The focus of this work is on characterization of uncertainties in distributed parameter systems. Two specific problem which represent the advection problem and the second which represents the diffusion problem are considered and analyzed. Closed form solutions for the deterministic problem are first developed for both the advection and diffusion problem. Following this, the evolution of uncertain initial conditions is studied. Dynamic models for the propagation of the mean and covariance of the advection and diffusion problem are derived and numerical simulations of the equations for the dynamics of the mean and variance are verified by comparing the results generated by Monte Carlo simulations. In the penultimate chapter, the effect of model parameter uncertainties is presented. A novel technique called Polynomial Chaos is used to parameterize the time-invariant uncertain model parameters. Galerkin projections are used to arrive at a set of coupled partial differential equations. Numerical simulation of this set of deterministic equations can be post-processed to determine the evolution of the statistics of the variable of interest. The advection and diffusion equations are again used to illustrate the polynomial chaos approach.