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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

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 2 , October 2009 , pp. 178 - 190
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Broadbridge, P. and Tritscher, P., “An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces”, IMA J. Appl. Math. 53 (1994) 249–265.CrossRefGoogle Scholar
[2]Caughey, D. A. and Hafez (eds), M. M., Frontiers of computational fluid dynamics (World Scientific, Singapore, 2006).Google Scholar
[3]Galaktionov, V., “On higher-order viscosity approximations of odd-order nonlinear PDEs”, J. Engng. Math. 60 (2008) 173–208.CrossRefGoogle Scholar
[4]Hinch, E. J., Perturbation methods (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[5]Kyrychko, Y. N., Bartuccelli, M. V. and Blyuss, K. B., “Persistence of travelling wave solutions of a fourth-order diffusion system”, J. Comput. Appl. Math. 176 (2005) 433–443.CrossRefGoogle Scholar
[6]Lagerstrom, P. A. and Casten, R. G., “Basic concepts underlying singular perturbation techniques”, SIAM Rev. 14 (1972) 63–120.CrossRefGoogle Scholar
[7]Leach, J. A. and Needham, D. J., Matched asymptotic expansions in reaction–diffusion theory (Springer, London, 2003).Google Scholar
[8]McCord, C. K., “Uniqueness of connecting orbits in the equation y (3)=y 2−1”, J. Math. Anal. Appl. 114 (1986) 584–592.CrossRefGoogle Scholar
[9]Nayfeh, A. H., Introduction to perturbation techniques (Wiley, New York, 1981).Google Scholar
[10]Van Dyke, M., Perturbation methods in fluid dynamics (Parabolic Press, Stanford, 1975).Google Scholar