Methods of Constructions of Exact Solutions to Nonlinear Reaction-Diffusion-Convection Equations

Abstract:
Nonlinear reaction-diffusion-convection (RDC) equations of the form

where T = T(t,x) is the unknown function and A(T), B(T),C(T) are arbitrary smooth functions, are considered. The equation (1) generalizes a number of the well known nonlinear second-order evolution equations, describing various processes in physics, chemistry, and biology.

Nowadays the most popular approaches for construction of exact solutions to nonlinear RDC equations of the form (1) are the classical Lie method and the Bluman-Cole approach of non-classical (i.e., Q-conditional) symmetries. Although the technique of both approaches is well known, new results are constantly obtained if non-trivial Lie or Q-conditional symmetries have been found for a given equation (1). Several other approaches for solving nonlinear evolution PDEs were independently suggested in middle 1990-ies (the method of additional generating conditions, the method of linear invariant subspaces, the method of generalized conditional symmetries, the method of heir-equations).

The talk is devoted to exhibition of main results obtained for RDC equation (1) using the classical Lie method, the Bluman-Cole approach and the method of additional generating conditions. The general scheme for solving nonlinear PDEs using all methods listed above is presented.