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The Quarterly Journal of Mechanics and Applied Mathematics 2006 59(4):631-650; doi:10.1093/qjmam/hbl019
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Q. Jl Mech. Appl. Math, Vol. 59. No. 4 © The author 2006. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Polynomial representations for initial-boundary-value problems involving the inviscid Proudman–Johnson equation

RE Grundy{dagger}

( School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews KY16 9SS )

{dagger} reg{at}st-andrews.ac.uk


  Abstract

The central aim of this paper is to show how two-point Hermite interpolation can be used to construct polynomial representations of solutions to some initial-boundary-value problems for the inviscid Proudman–Johnson equation. This classic equation of fluid dynamics can be regarded as first-order hyperbolic, and an important by-product of our analysis is an understanding of how Hermite interpolation can be utilized for such equations. Different types of boundary conditions may result in finite time blow-up and/or large time approach to the steady state depending on the value of a parameter appearing in the problem.


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