ABSTRACT
This chapter presents a survey of symmetries and exact solutions for diffusion equations. Such equations are widely used as mathematical models of heat conduction and diffusion, filtration of gases and fluids in porous media, processes of chemical kinetics and biological processes. A variety of applications of diffusion equations is given. The chapter also deals with one-dimensional equations. The linear heat equation admits Lie-Backlund symmetries of an arbitrary order. Symmetries allow one to find new solutions starting from any known solution as well as to construct invariant solutions. In order to enumerate all essentially different invariant solutions, it is necessary to use the optimal system of one-dimensional subalgebras. The expressions for the invariant solutions are written formally without any discussion of the range of variables and parameters where these expressions are meaningful.
