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Finding symmetries by incorporating initial conditions as side conditions

Published online by Cambridge University Press:  01 December 2008

JOANNA GOARD
JOANNA GOARD*
Affiliation:
University of Wollongong, School of Mathematics and Applied Statistics, Wollongong, NSW, Australia email: [email protected]
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Abstract

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 6 , December 2008 , pp. 701 - 715
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Bernoff, A. J. & Bertozzi, A. L. (1995) Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition. Physica D. 85, 375–404.CrossRefGoogle Scholar
[2]Bluman, G. W. & Cole, J. D. (1969) The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042.Google Scholar
[3]Bluman, G. W. & Kumei, S. (1989) Symmetries and Differential Equations, Springer-Verlag, New York.CrossRefGoogle Scholar
[4]Clarkson, P. A. & Kruskal, M. D. (1989) New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213.CrossRefGoogle Scholar
[5]Fokas, A. S & Liu, Q. M. (1994) Generalized conditional symmetries and exact solutions of nonintegrable equations. Theor. Math. Phys. 99, 263–277.CrossRefGoogle Scholar
[6]Goard, J. (2003) Noninvariant boundary conditions. Applicable Anal. 82 (5), 473–481.CrossRefGoogle Scholar
[7]Hill, J. M. (1992) Differential Equations and Group Methods for Scientists and Engineers. CRC Press, Boca Raton.Google Scholar
[8]Ibragimov, N. H. (1994/1995/1996) CRC Handbook of Lie Group Analysis of Differential Equations, Vols. 1–3, CRC Press, Boca Raton.Google Scholar
[9]Lie, S. (1881) Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch. Math. 6, 328–368.Google Scholar
[10]Olver, P. J. & Rosenau, P. (1986) The Construction of special solutions to partial differential equations. Phys. Lett. A. 114, 107–112.CrossRefGoogle Scholar
[11]Pucci, E. & Saccomandi, G. (1992) On the weak symmetry groups of partial differential equations. J. Math. Anal. Appl. 163, 588–598.CrossRefGoogle Scholar
[12]Pucci, E. & Saccomandi, G. (2000) Evolution equations, invariant surface conditions and functional separation of variables. Physica D. 139, 28–47.CrossRefGoogle Scholar
[13]Wilmott, P. (1998) Derivatives: The Theory and Practice of Financial Engineering, John Wiley and Sons Chichester, West Sussex.Google Scholar
[14]Zhdanov, R. Z. (1995) Conditional Lie-Bäcklund symmetry and reduction of evolution equations. J. Phys. A: Math. Gen. 28, 3841–3850.CrossRefGoogle Scholar