Enhanced preliminary group classification of a class of generalized diffusion equations

TL;DR

This paper advances the preliminary group classification method for generalized diffusion equations, providing a rigorous framework, overcoming common weaknesses, and thoroughly classifying symmetries for a specific class of equations.

Contribution

It introduces enhancements to the preliminary group classification method and applies it to a class of generalized diffusion equations, including a detailed symmetry classification.

Findings

Complete classification of inequivalent subalgebras

Extended the kernel algebra for the class of equations

Improved symmetry classification over previous work

Abstract

The method of preliminary group classification is rigorously defined, enhanced and related to the theory of group classification of differential equations. Typical weaknesses in papers on this method are discussed and strategies to overcome them are presented. The preliminary group classification of the class of generalized diffusion equations of the form u_t=f(x,u)u_x^2+g(x,u)u_{xx} is carried out. This includes a justification for applying this method to the given class, the simultaneous computation of the equivalence algebra and equivalence group, as well as the classification of inequivalent appropriate subalgebras of the whole infinite-dimensional equivalence algebra. The extensions of the kernel algebra, which are induced by such subalgebras, are exhaustively described. These results improve those recently published in Commun. Nonlinear Sci. Numer. Simul.