Using group theoretical methods, we analyze the generalization of a one-dimensional sixth-order thin filmequation which arises in considering the motion of a thin film of viscous fluid driven by an overlying elastic plate. The most general Lie group classification of point symmetries, its Lie algebra, and the equivalence group are obtained. Similarity reduction are performed and invariant solutions are constructed. It is found that some similarity solutions are of great physical interest such as sink and source solutions, travelling-wave solutions, waiting-time solutions, and blow-up solutions.
Received 14 July 2012Accepted 12 December 2012Published online 16 January 2013
This work was partially supported by the National Key Basic Research Project of China (Grant No. 2010CB126600), the National Natural Science Foundation of China (Grant No. 60873070), Shanghai Leading Academic Discipline Project No. B114, the Postdoctoral Science Foundation of China (Grant Nos. 20090450067 and 201104247), Shanghai Postdoctoral Science Foundation (Grant No. 09R21410600) and the Fundamental Research Funds for the Central Universities (Grant No. WM0911004).
Article outline: I. INTRODUCTION II. SYMMETRY CLASSIFICATION III. SIMILARITY REDUCTION A. f(u) is an arbitrary nonconstant function B. f(u) = eλu (λ ≠ 0) C. f(u) = um (m ≠ 0) IV. INVARIANT SOLUTIONS A. Source and sink solutions B. Travelling-wave solutions C. Waiting-time solutions D. Blow-up solutions V. CONCLUDING REMARKS
8.A. L. Bertozzi, “The mathematics of moving contact lines in thin liquid films,” Not. Am. Math. Soc.45, 689–697 (1998).
9.G. Bluman, A. Cheviakov, and S. Anco, Applications of Symmetry Methods to Partial Differential Equations (Springer-Verlag, Berlin, 2010).
10.G. W. Bluman and S. Kumei, Symmetries and Differential Equations (Springer-Verlag, Berlin, 1989).
11.M. S. Bruzon, M. L. Gandarias, E. Medina, and C. Muriel, “New symmetry reductions for a lubrication model,” in Proceedings of the Workshop Nonlinear Physics, Theory and Experiment, II: Università di Lecce-Consortium Einstein, 27 June-6 July, Gallipoli, Italy (World Scientific, Singapore, 2003).
12.P. Carbonaro, “Similarity solutions in one-dimensional relativistic gas dynamics,” Phys. Rev. E56(3), 2896–2902 (1997).
18.V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman and Hall/CRC, Boca Raton, FL, 2007).
19.M. L. Gandarias and M. S. Bruzon, “Symmetry analysis and solutions for a family of Cahn-Hilliard equations,” Rep. Math. Phys.46(1–2), 89–97 (2000).
24.K. D. Hobart, F. J. Kub, M. Fatemi, M. E. Twigg, P. E. Thompson, T. S. Kuan, and C. K. Inoki, “Compliant substrates: A comparative study of the relaxation mechanisms of strained films bonded to high and low viscosity oxides,” J. Electron. Mater.29(7), 897–900 (2000).
28.J. Hulshof, “Some aspects of the thin film equation, in European Congress of Mathematics,” Vol. II (Barcelona, 2000), Vol. 202 of Progr. Math., Birkhäuser, Basel, 2001, pp. 291–301.
29.N. H. Ibragimov, Lie Group Analysis of Differential Equations – Symmetries, Exact Solutions, and Conservation Laws, V. 1 (CRC, Boca Raton, FL, 1994).
30.N. M. Ivanova, R. O. Popovych, and C. Sophocleous, “Group analysis of variable coefficient diffusion-convection equations. I. Enhanced group classification,” Lobachevskii J. Math.31(2), 100–122 (2010).