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We study the application of generalized symmetry for reducing nonlinear partial differential equations. We construct the ansatzes for dependent variable u which reduce the scalar partial differential equation with two independent variables to systems of ordinary differential equations. The operators of Lie-Bäcklund symmetry of the second order ordinary differential equation are used. We apply the method to nonlinear evolutionary equations and find solutions which cannot be obtained in the framework of classical Lie approach. The method is also applicable to partial differential equations which are not restricted to evolution type ones. We construct the solution of nonlinear hyperbolic equation depending on an arbitrary smooth function on one variable. We study also the correlation between the dimension of symmetry Lie algebra and possibility of constructing non-invariant solutions to the equation under study.

We study the symmetry reduction of nonlinear partial differential equations which are used for describing diffusion processes in nonhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary differential equations. The ansatzes are constructed by using operators of Lie-Bäcklund symmetry of third order ordinary differential equation. The method gives the possibility to find solutions which cannot be obtained by virtue of classical Lie method. Such solutions have been constructed for nonlinear diffusion equations which are invariant with respect to one-parameter, two-parameter and three-parameter Lie groups of point transformations.

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The airflow in the room or any space has a significant impact on the proper operation of the entire ventilation system. An appropriate field of air velocity flowing through the entire room allows for climate comfort. That is why it is so important proper selection as well as proper arrangement for the air-supply and air-exhaust elements in the ventilation system. This research study presents an isothermal air stream flowing from two different slot diffusers. The supply airstream was analysed at various volumetric flow rates and the velocity of the supply air stream was measured in accordance with the PN-EN 12238: 2002 standard. In order to evaluate the optimal measuring points and to create an appropriate grid, before starting the experimental measurement smoke tests were carried out which determine the scope and shape of the supply stream. All experimental measurements were performed up-to the location where the supply air stream reaches the limiting speed of 0.2 m/s. As a result of the analysis, the obtained characteristics of two types of slot diffusers were discussed and compared.