The Large-Time Solution of a Nonlinear Fourth-Order Equation Initial-Value Problem I. Initial Data With a Discontinuous Expansive Step

John Leach, AP Bassom

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Abstract

In this paper we consider an initial-value problem for the nonlinear fourth-order partial differential equation u(t) + uu(x) +. gamma u(xxxx) = 0, -infinity <x <infinity, t > 0, where x and t represent dimensionless distance and time respectively and. is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that u(x, 0) = u(0)( > 0) for x >= 0 and u(x, 0) = 0 for x <0. The method of matched asymptotic expansions is used to obtain the large-time asymptotic structure of the solution to this problem which exhibits the formation of an expansion wave. Whilst most physical applications of this type of equation have gamma > 0, our calculations show how it is possible to infer the large-time structure of a whole family of solutions for a range of related equations.
Original languageEnglish
Pages (from-to)178-190
Number of pages13
JournalThe ANZIAM Journal
Volume51
Issue number02
DOIs
Publication statusPublished - 1 Oct 2009

Keywords

  • asymptotic analysis
  • partial differential equation

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