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07 Nov 2012
Symmetry properties for a generalised thin film equation
 Kyriakos Charalambous,
 Christodoulos Sophocleous
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Symmetry properties are presented for a fourthorder parabolic equation written in conservation form. It was introduced in the literature as a generalisation of the fourthorder thin film equation. We derive equivalence transformations, Lie symmetries, potential symmetries, nonclassical symmetries and potential nonclassical symmetries. A chain of such equations is introduced. We conclude by presenting similar results for the thirdorder equation of this chain.
 Myers, TG (1998) Thin films with high surface tension. SIAM Rev 40: pp. 441462 CrossRef
 King, JR (2001) Two generalisations of the thin film equation. Math Comput Model 34: pp. 737756 CrossRef
 Bertozzi, AL, Pugh, M (1998) Longwave instabilities and saturation in thin film equations. Commun Pure Appl Math 51: pp. 625661 CrossRef
 Constantin, P, Dupont, TF, Goldstein, PE, Kadanoff, LP, Shelley, MJ, Zhou, SM (1993) Droplet breakup in a model of the HeleShaw cell. Phys Rev E 47: pp. 41694181 CrossRef
 Hocherman, T, Rosenau, P (1993) On KStype equations describing the evolution and rupture of a liquid interface. Physica D 67: pp. 113125 CrossRef
 King, JR, Bowen, M (2001) Moving boundary problem and nonuniqueness for thin film equations. Eur J Appl Math 7: pp. 321356
 Smyth, NF, Hill, JM (1988) Highorder nonlinear diffusion. IMA J Appl Math 40: pp. 7386 CrossRef
 Sophocleous, C, Leach, PGL (2012) Thin films: increasing the complexity of the model. Int J Bifur Chaos Appl Sci Eng 22: pp. 1250212 CrossRef
 Ibragimov, NH eds. (1994) Symmetries, exact solutions and conservation laws: Lie group analysis of differential equations, vol 1. CRC, Boca Raton, FL
 Ibragimov, NH (eds) (1995) Applications in engineering and physical sciences: Lie group analysis of differential equations, vol 2. CRC, Boca Raton, FL
 Ibragimov, NH eds. (1996) New trends in theoretical developments and computational methods: Lie group analysis of differential equations, vol 3. CRC, Boca Raton, FL
 Ovsiannikov LV (1959) Group relations of the equation of nonlinear heat conductivity. Dokl Akad Nauk SSSR 125:492–495 (in Russian)
 Ivanova, NM, Popovych, RO, Sophocleous, C (2010) Group analysis of variable coefficient diffusion–convection equations. I: Enhanced group classification. Lobachevskii J Math 31: pp. 100122 CrossRef
 Vaneeva, OO, Popovych, RO, Sophocleous, C (2009) Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source. Acta Appl Math 106: pp. 146 CrossRef
 Vaneeva, OO, Johnpillai, AG, Popovych, RO, Sophocleous, C (2007) Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities. J Math Anal Appl 330: pp. 13631386 CrossRef
 Bluman, GW, Kumei, S (1989) Symmetries and differential equations. Springer, New York CrossRef
 Arrigo, DJ, Hill, JM (1995) Nonclassical symmetries for nonlinear diffusion and absorption. Stud Appl Math 94: pp. 2139
 Gandarias, ML (2001) New symmetries for a model of fast diffusion. Phys Lett A 286: pp. 153160 CrossRef
 Popovych, RO, Vaneeva, OO, Ivanova, NM (2007) Potential nonclassical symmetries and solutions of fast diffusion equation. Phys Lett A 362: pp. 166173 CrossRef
 Lie S (1884) Klassifikation und Integration von gewohnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten IV. Archiv for Matematik og Naturvidenskab 9:431448. Reprinted in Lie’s Ges. Abhandl. 5, paper XVI, 1924, 432–446
 Ovsiannikov, LV (1982) Group analysis of differential equations. Academic, New York
 Kingston, JG, Sophocleous, C (1998) On formpreserving point transformations of partial differential equations. J Phys A 31: pp. 15971619 CrossRef
 Popovych, RO, Ivanova, NM (2004) New results on group classification of nonlinear diffusion–convection equations. J Phys A 37: pp. 75477565 CrossRef
 Ibragimov, NH (2004) Equivalence groups and invariants of linear and nonlinear equations. Arch ALGA 1: pp. 969
 Popovych, RO, Ivanova, NM (2005) Potential equivalence transformations for nonlinear diffusionconvection equations. J Phys A 38: pp. 31453155 CrossRef
 Sophocleous, C (1999) Continuous and discrete transformations of a onedimensional porous medium equation. J Nonlinear Math Phys 6: pp. 355364 CrossRef
 Kingston, JG, Sophocleous, C (1991) On point transformations of a generalised Burgers equation. Phys Lett A 155: pp. 1519 CrossRef
 Popovych, RO, Kunzinger, M, Eshraghi, H (2010) Admissible transformations and normalized classes of nonlinear Schrödinger equations. Acta Appl Math 109: pp. 315359 CrossRef
 Bluman, GW, Anco, SC (2002) Symmetry and integration methods for differential equations. Springer, New York
 Bluman, GW, Cheviakov, AF, Anco, SC (2010) Applications of symmetry methods to partial differential equations. Springer, New York CrossRef
 Ibragimov, NH (1999) Elementary Lie group analysis and ordinary differential equations. Wiley, New York
 Olver, P (1986) Applications of Lie groups to differential equations. Springer, New York CrossRef
 Bluman, GW, Reid, GJ, Kumei, S (1988) New classes of symmetries for partial differential equations. J Math Phys 29: pp. 806811 CrossRef
 Popovych, RO, Seryeyev, A (2010) Consrvation laws and normal forms of evolution equations. Phys Lett A 374: pp. 22102217 CrossRef
 Ivanova, NM, Popovych, RO, Sophocleous, C, Vaneeva, OO (2009) Conservation laws and hierarchies of potential symmetries for certain diffusion equations. Physica A 388: pp. 343356 CrossRef
 Sophocleous, C (2005) Further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients. Physica A 345: pp. 457471
 Bluman, GW, Cole, JD (1969) The general similarity solution of the heat equation. J Math Mech 18: pp. 10251042
 Fushchich, WI, Tsyfra, IM (1987) On a reduction and solutions of the nonlinear wave equations with broken symmetry. J Phys A 20: pp. L45L48 CrossRef
 Zhdanov, RZ, Tsyfra, IM, Popovych, RO (1999) A precise definition of reduction of partial differential equations. J Math Anal Appl 238: pp. 101123 CrossRef
 Fushchich WI, Serov NI (1988) Conditional invariance and exact solutions of a nonlinear acoustics equation. Dokl Akad Nauk Ukrain SSR A 10:2731 (in Russian)
 Fushchich, WI, Shtelen, WM, Serov, MI, Popovych, RO (1992) Qconditional symmetry of the linear heat equation. Proc Acad Sci Ukraine 12: pp. 2833
 Levi, D, Winternitz, P (1989) Nonclassical symmetry reduction: example of the Boussinesq equation. J Phys A 22: pp. 29152924 CrossRef
 Polyanin, AD, Zaitsev, VF (2003) Handbook of exact solutions for ordinary differential equations. Chapman & Hall/CRC, Boca Raton
 Oron, A, Rosenau, P (1986) Some symmetries of the nonlinear heat and wave equations. Phys Lett A 118: pp. 172176 CrossRef
 Calogero, F Why are certain partial differential equations both widely applicable and integrable?. In: Zakharov, VE eds. (1991) What is integrability? Springer series in nonlinear dynamics. Springer, New York
 Title
 Symmetry properties for a generalised thin film equation
 Journal

Journal of Engineering Mathematics
Volume 82, Issue 1 , pp 109124
 Cover Date
 20131001
 DOI
 10.1007/s1066501295776
 Print ISSN
 00220833
 Online ISSN
 15732703
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Lie symmetries
 NonLie reductions
 Potential symmetries
 Thin film equations
 Industry Sectors
 Authors

 Kyriakos Charalambous ^{(1)}
 Christodoulos Sophocleous ^{(1)}
 Author Affiliations

 1. Department of Mathematics and Statistics, University of Cyprus, Nicosia, 1678, Cyprus