Kovalenko S. Lie symmetries and exact solutions some nonlinear boundary value problems with moving boundaries

The thesis is devoted to development and improvement of the methods of theoretic-group analysis of boundary value problems and their applications to investigation of the symmetry properties for several nonlinear boundary value problems of the Stefan type. Exact invariant solutions of the boundary value problems involved are constructed and their physical interpretation is also proposed. A new definition of Lie invariance for a boundary value problem with boundary conditions at fixed and moving boundaries, and boundary conditions at infinity is formulated. The problem of group classification of classes of the boundary value problems is suggested, and an algorithm for solution of such a problem is worked out. The group classification of several classes of nonlinear boundary value problems of the Stefan type modelling the processes of melting and evaporation of materials, which are exposed under the action of power energy fluxes are carried out. Results of the classification obtained are applied to construction of new exact invariant solutions of some boundary value problems from the class under study. The physical interpretation for each solution obtained is proposed.