# Polynomial Chaos Methods for Hyperbolic Partial Differential EquationsBoundary Conditions and Data

Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Boundary Conditions and Data [We continue analysis of Burgers’ equation from the previous chapter with a focus on the effect of data for the boundaryboundary conditions conditions. To facilitate understanding, we deal only with the truncated representation u(x,t,ξ)=u0ψ0+u1ψ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u(x,t,\xi ) = u_{0}\psi _{0} + u_{1}\psi _{1}$$ \end{document}. This means that all the stochastic variation is accounted for by the single gPC coefficient u1, and the standard deviation of the solution is simply | u1 | . 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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Polynomial Chaos Methods for Hyperbolic Partial Differential EquationsBoundary Conditions and Data

Part of the Mathematical Engineering Book Series
11 pages

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# References (1)

Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2015
ISBN
978-3-319-10713-4
Pages
111 –121
DOI
10.1007/978-3-319-10714-1_7
Publisher site
See Chapter on Publisher Site

### Abstract

[We continue analysis of Burgers’ equation from the previous chapter with a focus on the effect of data for the boundaryboundary conditions conditions. To facilitate understanding, we deal only with the truncated representation u(x,t,ξ)=u0ψ0+u1ψ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u(x,t,\xi ) = u_{0}\psi _{0} + u_{1}\psi _{1}$$ \end{document}. This means that all the stochastic variation is accounted for by the single gPC coefficient u1, and the standard deviation of the solution is simply | u1 | . 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.]

Published: Sep 17, 2014

Keywords: Unknown Standard Deviation; Boundary Data; Unknown Expectation; Simple Apparatus; Stochastic Variables

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