Contractions of Lie algebras are combined with the classical matrix method of Gel’fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudounitary Lie algebras Iu(p, q). This procedure is extended to contractions of Iu(p, q) isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Li...

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the [Formula: see text]-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponen...

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elast...

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie’s method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity ...