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 language | English |
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Pages (from-to) | 178-190 |
Number of pages | 13 |
Journal | The ANZIAM Journal |
Volume | 51 |
Issue number | 02 |
DOIs | |
Publication status | Published - 1 Oct 2009 |
Keywords
- asymptotic analysis
- partial differential equation