ABSTRACT
Chapter 3 is devoted to the direct method of symmetry reductions and the direct method of weak symmetry reductions. The chapter opens with a simplified scheme of the former and discusses special forms of reductions to facilitate understanding of the method. The general form of reductions is further considered. The direct method of weak symmetry reductions combines the former with the method of functional separation of variables. Unlike the direct method of symmetry reductions, using weak symmetry reductions assumes that the outer function can satisfy an overdetermined system of several ordinary differential equations rather than a single equation. This approach often allows one to obtain significantly more solutions, many of which can be represented in a closed form. The application of the methods is illustrated with numerous examples and tables, and the focus is, as before, on nonlinear equations of hydrodynamics, heat and mass transfer, and wave theory.
