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A differential constraints approach to partial invariance

Published online by Cambridge University Press:  26 September 2008

Jeffrey Ondich
Affiliation:
Department of Mathematics and Computer Science, Carleton College, Northfield, MN 55057, USA

Abstract

Ovsiannikov's method of partially invariant solutions of differential equations can be considered to be a special case of the method of differential constraints introduced by Yanenko and by Olver and Rosenau. Differential constraints are used to construct non-reducible partially invariant solutions of the boundary layer or Prandtl equations.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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References

[1] Yanenko, N. N. 1964 Theory of consistency and methods of integrating systems of nonlinear partial differential equations. Proc. Fourth All-Union Mathematics Congress, pp. 247259. Nauka, Leningrad (in Russian).Google Scholar
[2] Sidorov, A. F., Shapeev, V P. & Yanenko, N. N. 1984 The method of differential constraints and its applications in gas dynamics. Nauka, Novosibirsk (in Russian).Google Scholar
[3] Olver, P. J. & Rosenau, P. 1986 The construction of special solutions to partial differential equations. Physics Letters 114A: 107112.CrossRefGoogle Scholar
[4] Olver, P. J. 1986 Applications of Lie Groups to Differential Equations. Springer-Verlag, New York.CrossRefGoogle Scholar
[5] Ovsiannikov, L. V. 1982 Group Analysis of Differential Equations. Academic Press, New York.Google Scholar
[6] Bluman, G. W. & Cole, J. D. 1969 The general similarity solution of the heat equation. J. Math. Mech. 18: 10251042.Google Scholar
[7] Olver, P. J. & Rosenau, P. 1987 Group-invariant solutions of differential equations. SIAM J. Appl. Math. 47: 263278.CrossRefGoogle Scholar
[8] Ondich, J. 1995 The reducibility of partially invariant solutions of systems of partial differential equations. Euro. J. Appl. Math. 6 (4): 329354.CrossRefGoogle Scholar
[9] Ovsiannikov, L. V. 1962 Group Properties of Differential Equations. Novosibirsk, Moscow (in Russian).Google Scholar
[10] Pukhnachov, V. V. 1984 Group analysis of the equations of the nonstationary Marangoni boundary layer. Soviet Physics Doklady 29 (12): 978980.Google Scholar
[11] Kamke, E. 1971 Differentialgleichungen Lösungsmethoden unci Lösungen. Chelsea, New York.Google Scholar
[12] Vorob'ev, E. M. 1991 Reduction and quotient equations for differential equations with symmetries. Acta Applicandae Mathematicae 23: 124.CrossRefGoogle Scholar
[13] Martina, L. & Winternitz, P. 1992 Partially invariant solutions of nonlinear Klein-Gordon and Laplace equations. J. Math. Phys. 33: 27182727.CrossRefGoogle Scholar
[14] Martina, L., Soliani, G. & Winternitz, P. 1992 Partially invariant solutions of a class of nonlinear Schrodinger equations. J. Phys. A: Math. Gen. 25: 44254435.CrossRefGoogle Scholar