On reducing and finding solutions of nonlinear evolutionary equations via generalized symmetry of ordinary differential equations.

We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.