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.
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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
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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
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