# Groupe de Physique Statistique

## Equipe 106, Institut Jean Lamour

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### Séminaire de groupe

 Symmetries and exact solutions of boundary-value problems: new definitions and applications. Roman Cherniha Dept. des Mathematiques, Nottingham (Angleterre) et Academie Ukraineenne des Sciences, Kyiv (Ukraine) mardi 17 mars 2015 , 14h15 Salle de séminaire du groupe de Physique Statistique Nowadays, Lie symmetries are widely applied to study nonlinear partial differential equations (including multidimensional PDEs), notably for their reductions to ordinary differential equations and constructing exact solutions. There are a huge number of papers and many excellent books devoted to such applications. Over recent decades, other symmetry methods, which are based on the classical Lie method, were also derived and applied for solving nonlinear PDEs. On the other hand, one may note that the symmetrybased methods were not widely used for solving boundary-value problems (BVPs). The first rigorous definition of Lie’s invariance for BVPs was formulated by G. Bluman in early 1970s and applied to some two-dimensional BVPs based on the linear heat equation. However, Bluman’s definition cannot be directly applied to BVPs of more general form, for example, to those involving boundary conditions on the moving surfaces and at infinity.
Recently, we formulated new definitions of Lie and conditional invariance of multidimensional BVPs involving a wide range of boundary conditions (including those at infinity and moving surfaces). An algorithm for finding such symmetries for the given class of BVPs was worked out. In order to show their efficiency, the definitions and algorithm were applied to some classes of nonlinear BVPs (including multidimensional problems and those with moving boundaries) arising in physical and biological applications. As a result, the Lie and conditional symmetries for these BVPs were completely described, examples of reductions to BVPs of lower dimensionality and exact solutions were constructed. Some physical interpretation of the results obtained is also presented.
[1] Cherniha R and Kovalenko S, J. Phys. A 44, 485202 (2011)
[2] Cherniha R and Kovalenko S, Comm.. Nonl. Sci. Num.. Simul. 17, 71 (2012)
[3] Cherniha R. and King J. R., arxiv:1412.6967 (2014).