Boundary Conditions and Data

Abstract

We continue analysis of Burgers’ equation from the previous chapter with a focus on the effect of data for the boundary conditions. To facilitate understanding, we deal only with the truncated representation $u(x,t,\xi ) = u_{0}\psi _{0} + u_{1}\psi _{1}$. This means that all the stochastic variation is accounted for by the single gPC coefficient u ₁, and the standard deviation of the solution is simply | u ₁ | . With this simplified setup, we obtain a few combinations of general situations for the boundary data: known expectation but unknown standard deviation, unknown expectation and standard deviation, etc. The implication in all these situations on well-posedness, stability and accuracy is discussed.