Abstract
The first chapter is a brief, but a sufficiently comprehensive introduction to the methods of Lie group analysis of ordinary and partial differential equations. The chapter presents basic concepts from the theory: continuous transformation groups, their generators, Lie equations, groups admitted by differential equations, integration of ordinary differential equations using their symmetries, group classification and invariant solutions of partial differential equations. New trends in modern group analysis such as the theory of Lie–Bäcklund transformations groups and approximate groups are also reflected. The intention of the chapter is to give the basic ideas of classical and modern group analysis to beginner readers and provide useful materials for advanced specialists.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Manifolds are treated locally and all functions under consideration are supposed to be continuous and differentiable sufficiently many times.
- 2.
In [12], Sect. III, Lie proves a more general statement about contact transformations of parabolic equations.
References
Bäcklund, A.V.: Ueber Flächentransformationen. Math. Ann. IX, 297–320 (1876)
Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Approximate symmetries. Math. Sb. 136(178)(3), 435–450 (1988). English transl.: Math. USSR Sb. 64(2), (1989)
Ibragimov, N.H.: Sur l’équivalence des équations d’évolution, qui admettent une algébre de Lie–Bäcklund infinie. C. R. Acad. Sci. Paris Sér. I 293, 657–660 (1981)
Ibragimov, N.H.: Transformation Groups in Mathematical Physics. Nauka, Moscow (1983). English transl.: Transformation Groups Applied to Mathematical Physics. Riedel, Dordrecht (1985)
Ibragimov, N.H.: Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie). Usp. Mat. Nauk 47(4), 83–144 (1992) English transl.: Russ. Math. Surv. 47(2), 89–156 (1992)
Ibragimov, N.H. (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2: Applications in Engineering and Physical Sciences. CRC Press, Boca Raton (1995)
Ibragimov, N.H. (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3: New Trends in Theoretical Developments and Computational Methods. CRC Press, Boca Raton (1996)
Ibragimov, N.H.: Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley, Chichester (1999)
Ibragimov, N.H. (ed.): Lie Group Analysis: Classical Heritage. ALGA Publications, Karlskrona (2004)
Ibragimov, N.H.: A Practical Course in Differential Equations and Mathematical Modelling, 3rd edn. ALGA Publications, Karlskrona (2006)
Ibragimov, N.H.: Optimal system of invariant solutions for the Burgers equation. In: Lecture at the MOGRAN-12 Conference, Porto, July 28–31, 2008
Lie, S.: Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch. Math. 6(3), 328–368 (1881). English transl. in: Ibragimov, N.H. (ed.) CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2: Applications in Engineering and Physical Sciences. CRC Press, Boca Raton (1995). Reprinted also in: Ibragimov, N.H. (ed.) Lie Group Analysis: Classical Heritage. ALGA Publications, Karlskrona (2004)
Lie, S.: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen. Bearbeited und herausgegeben von Dr. G. Scheffers. Teubner, Leipzig (1891)
Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, San Diego (1974)
Olver, P.J.: Applications of Lie Groups to Differential Equations, Springer, New York (1986). 2nd edn. (1993)
Ovsyannikov, L.V.: Group Properties of Differential Equations. Siberian Branch, USSR Academy of Sciences, Novosibirsk (1962) (in Russian)
Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V. (2010). Introduction to Group Analysis of Differential Equations. In: Symmetries of Integro-Differential Equations. Lecture Notes in Physics, vol 806. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3797-8_1
Download citation
DOI: https://doi.org/10.1007/978-90-481-3797-8_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3796-1
Online ISBN: 978-90-481-3797-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)