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Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

Published online by Cambridge University Press:  16 July 2009

G. W. Bluman  and
S. Kumei
G. W. Bluman
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, CanadaV6T 1Y4
S. Kumei
Affiliation:
Faculty of Textile Science and Technology, Shinshu University, Ueda, Nagano 386, Japan
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Abstract

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 3 , September 1990 , pp. 189 - 216
Copyright
Copyright © Cambridge University Press 1990

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References

Bäcklund, A. V. 1876 Ueber Flächentransformationen. Math. Ann. 9, 297–320.Google Scholar
Bluman, G. W. 1974 Use of group methods for relating linear and nonlinear partial differential equations. Proc. Symp. on Symmetry, Similarity and Group Theoretic Methods in Mechanics, Calgary, pp.203–218.Google Scholar
Bluman, G. W. 1980 On the transfonnation of diffusion processes into the Wiener process. SIAM J. Appl. Math. 39, 238–247.Google Scholar
Bluman, G. W. 1983 On mapping partial differential equations to constant coefficient equations. SIAM J. Appl. Math. 43, 1259–1273.Google Scholar
Erratum: Zentralblatt für Math. 544.35007 (1985).Google Scholar
Bluman, G. W. 1990 Simplifying the form of Lie groups admitted by a given differential equation. J. Math. Anal. Appl. 145, 52–62.Google Scholar
Bluman, G. W. & Cole, J. D. 1974 Similarity Methods for Differential Equations. Appl. Math. Sci. No. 13, Springer-Verlag, New York.CrossRefGoogle Scholar
Bluman, G. W. & Kumei, S. 1989 Symmetries and Differential Equations. Appl. Math. Sci. No. 81, Springer-Verlag, New York.Google Scholar
Bluman, G. W., Kumei, S. & Reid, G. J. 1988 New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811.CrossRefGoogle Scholar
Erratum: J. Math. Phys. 29, 2320 (1988).Google Scholar
Forsyth, A. R. 1906 Theory of Dfferential Equations. Vol. v., Cambridge University Press, p. 318.Google Scholar
Korobeinkov, V. P. 1982 Certain types of solutions of Korteweg-de Vries-Burgers' equations for plane, cylindrical, and spherical waves, Proc. IUTAM Symp. on Nonlinear Wave Deformations, Tallinn (in Russian).CrossRefGoogle Scholar
Kumei, S. & Bluman, G. W. 1982 When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173.Google Scholar
Lie, S. 1890 Theorie der Transformationsgruppen. Vol. II, B. G. Teubner, Leipzig.Google Scholar
Mayer, A. 1875 Directe Begrührundung der Theorie der Beruhrungstransformationen. Math. Ann. 8, 304–312.Google Scholar
Miura, R. M. 1968 Korteweg-de Vries equation and generalizations: I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202–1204.Google Scholar
Müller, E. A. & Matschat, K. 1962 Über das Auffinden von Ähnlichkeitslösungen partieller Differentialgleichungssysteme ünter Benutzung von Transformationsgruppen, mit Anwendungen auf Probleme der Strömungsphysik, Miszellaneen der Angewandten Mechanik, Berlin 190–222.Google Scholar
Olver, P. J. 1986 Applications of Lie Groups to Differential Equations. GTM, No. 107, Springer- Verlag, New York.Google Scholar
Ovsiannikov, L. V. 1982 Group Analysis of Differential Equations. Academic Press, New York.Google Scholar
Reid, G. J. 1990 Finding symmetries of differential equations without integrating determining equations. IAM Technical Report No. 90–4, University of British Columbia.Google Scholar
Sukharev, M. G. 1967 Invariant solutions of equations describing the motion of fluids and gases in long pipelines. Dokl. Akad. Nauk USSR 175, 781–784 (in Russian).Google Scholar
Varley, E. & Seymour, B. 1985 Exact solutions for large amplitude waves in dispersive and dissipative systems. Stud. Appl. Math. 72, 241–262.Google Scholar