Some Questions For The Perturbed Nonlinear Diffusion-convection Equations

The nonlinear phenomena widely appears in almost all the scientific fields such as mathematics, physics, chemistry, biology and so on. With the development of science, man studying on nonlinear systems has been making more and more progress. As a result, to deal with the nonlinear equations that describe the nonlinear systems becomes one of the main and hot topics for the researchers. As we well know, the symmetry reduction methods related to group theory, such as the Lie point symmetry method, conditional symmetry, generalized conditional symmetry and etc, are effective tools to study the exact solutions of the nonlinear partial differential equations.Meanwhile, many new perturbed PDEs in science and technology need us to seek out approximate solutions. Recently, several symmetry-based perturbation methods have been developed to deal with the perturbed PDEs.This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry(AGCS). The following topics have been concerned:(1) The concept of approximate generalized conditional symmetry(AGCS) for the perturbed evolution equations as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced;(2) Complete classification of the perturbed nonlinear diffusion-convection equations which admit certain types of AGCSs is derived;(3) Some approximate solutions to some perturbed equations can be obtained via the AGCSs.