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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

journal contribution
posted on 2023-05-18, 15:45 authored by Leach, JA, Andrew BassomAndrew Bassom
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.

History

Publication title

The ANZIAM Journal

Volume

51

Pagination

178-190

ISSN

1446-1811

Department/School

School of Natural Sciences

Publisher

Australian Mathematics Publ Assoc Inc

Place of publication

Mathematics Dept Australian National Univ, Canberra, Australia, Act, 0200

Rights statement

Copyright Australian Mathematical Society 2010

Repository Status

  • Restricted

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