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THE LARGE-TIME SOLUTION OF A NONLINEAR FOURTH-ORDER EQUATION INITIAL-VALUE PROBLEM I. INITIAL DATA WITH A DISCONTINUOUS EXPANSIVE STEP

Part of: Partial differential equations

Published online by Cambridge University Press:  18 May 2010

J. A. LEACH  and
ANDREW P. BASSOM
J. A. LEACH*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (email: leachja@for.mat.bham.ac.uk)
ANDREW P. BASSOM
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley 6009, Australia (email: bassom@maths.uwa.edu.au)
*
✉For correspondence; e-mail: leachja@for.mat.bham.ac.uk
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Abstract

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In this paper we consider an initial-value problem for the nonlinear fourth-order partial differential equation ut+uux+γuxxxx=0, −∞<x<∞, t>0, where x and t represent dimensionless distance and time respectively and γ is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that u(x,0)=u0(>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 γ>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.

Keywords

partial differential equationasymptotic analysis

MSC classification

Secondary: 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 2 , October 2009 , pp. 178 - 190
Copyright
Copyright © Australian Mathematical Society 2010

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