New results on group classification of nonlinear diffusion-convection equations

TL;DR

This paper introduces a new method for classifying nonlinear diffusion-convection equations with variable coefficients, identifying equations with large symmetry groups, and constructing exact solutions using advanced transformation techniques.

Contribution

The authors develop a novel approach utilizing conditional and partial equivalence transformations for classifying a broad class of nonlinear diffusion-convection equations.

Findings

Identified new equations with large invariance algebras

Constructed exact solutions for specific cases

Described all possible partial equivalence transformations

Abstract

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.