Popovych R. Classification problems of group analysis of differential equations

The thesis is devoted to the enhancement of existing methods and the development of new methods for group analysis of differential equations and related areas of the theory of Lie algebras. In particular, a number of new concepts regarding classes of differential equations are introduced: an extended (resp. generalized extended, resp. conditional, resp. potential) equivalence group, a normalized (resp. semi-normalized, resp. strongly normalized) class, similar classes, a mapping between classes which is generated by a family of point transformations, etc. Effective new techniques of group classification are introduced. They involve partitioning into normalized subclasses, branched splitting and mappings between classes. The applicability range of the algebraic method is determined. Classification problems for admissible transformations, Lie symmetries, conservation laws, potential symmetries and reduction operators with respect to different kinds of equivalences are rigorously posed and solved for a number of classes of differential equations important for applications. Conservation laws of foliated systems of differential equations and singular reduction operators for partial differential equations in two independent variables are studied. New necessary criteria for contractions of Lie algebras are obtained. In order to calculate invariants of Lie algebras, an original algorithm based on Cartan's method of moving frames is developed.