Symmetries and solutions to the thin film equations
type: Item
instance of: scholarly article
title: Symmetries and solutions to the thin film equations
zbMATH Open document ID: 1091.35004
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-04-06T00:00:00Z
review text: The author considers the determination of solutions by symmetry techniques of the nonlinear \(1 +1 \) fourth-order partial differential equation \[ u_t = -f (u)u_{xxxx} - g (u)u_xu_{xxx} - h (u)u_{xx} ^2 - d (u)u_x ^2u_{xx} + p (u)u_{xx} + q (u)u_x ^2 \tag{1} \] for certain smooth functions \(f \), \(g \), \(h \), \(d \), \(p \) and \(q \). Equation (1) for specific coefficient functions arises in the modelling of thin films and other fluid-mechanical applications. The author approaches the symmetry analysis of (1) with the assumption of specific classes of symmetry rather than the method of preliminary group classification which would seem to be a more natural approach since the assumption of a particular symmetry is more a matter of the taste of the assumer rather than the dictate of the nature of the equation. However, the author does make an interesting extension to the standard assumptions by the use of nonlinear extensions of the scaling group as introduced by Galaktionov. Consequently a variety of closed-formed solutions of (1) is obtained. Of the many cases considered some display divergence in finite time. It is probable that there are other classes of functions in (1) for which closed-formed solutions can be obtained by means of symmetry analysis. It would be interesting to see a singularity analysis would lead to different classes of solutions for this could give a pointer to further Ansätze for the proper choice of symmetries. The reader is advised that in the same context prime is used to denote differentiation with respect to different variables.
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 76A20
Mathematics Subject Classification ID: 74K35
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 5017388
zbMATH Keywords: Lie symmetries
zbMATH Keywords: similarity solutions
zbMATH Keywords: rotation group
zbMATH Keywords: scaling group
author: Changzheng Qu
reviewed by: P. G. L. Leach
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.07.040
OpenAlex ID: W2061920758
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DOI: 10.1016/J.JMAA.2005.07.040
description: scientific article; zbMATH DE number 5017388