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Symmetries of Boundary Value Problems: Definitions, Algorithms and Applications to Physically Motivated Problems

Cherniha, R (University of Nottingham)
Wednesday 25 June 2014, 11:15-11:45

Seminar Room 1, Newton Institute


One may note that the symmetry-based methods were not widely used for solving boundary-value problems (BVPs). To the best of our knowledge, the first rigorous definition of Lie's invariance for BVPs was formulated by George Bluman in early 1970s and applied to some classical BVPs. 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, which are described by unknown functions. In our recent papers, a new definition of Lie's invariance of BVP with a wide range of boundary conditions (including those at infinity and moving surfaces) was formulated. Moreover, an algorithm of the group classification for the given class of BVPs was worked out. The definition and algorithm were applied to some classes of nonlinear two-dimensional and multidimensional BVPs of Stefan type with the aim to show their efficiency. In particular, the group classification problem for these classes of BVPs was solved, reductions to BVPs of lower dimensionality were constructed and examples of exact solutions with physical meaning were found. Very recently, the definition and algorithms were extended on the case of conditional invariance for BVPs and applied to some nonlinear BVPs. This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.


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