Mapping method of group classification

TL;DR

This paper develops a generalized mapping method for group classification of differential equations, applying it to Kolmogorov and Fokker-Planck equations, and reduces classification problems to solving heat equations with potentials.

Contribution

It introduces the notion of weakly similar classes and generalizes the mapping method for group classification, enabling exhaustive solutions for specific classes.

Findings

Computed equivalence groupoids and groups for the classes

Reduced classification problems to heat equations with potentials

Provided exhaustive classification for time-independent coefficient cases

Abstract

We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes, which generalizes all the versions of this method that have been presented in the literature. The mapping method is applied to group classification of various classes of Kolmogorov equations and of Fokker-Planck equations in the case of space dimension one. The equivalence groupoids and the equivalence groups of these classes are computed. The group classification problems for these classes with respect to the corresponding equivalence groups are reduced to finding all inequivalent solutions of heat equations with inequivalent potentials admitting Lie-symmetry extensions. This reduction allows us to exhaustively solve the group classification problems for the classes of Kolmogorov and Fokker-Planck equations with time-independent coefficients.