Mapping method of group classification

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.

42 pages, 2 tables, minor revision

References in corpus (6)

• Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
• Conservation Laws and Potential Symmetries of Linear Parabolic Equations
• Enhanced preliminary group classification of a class of generalized diffusion equations
• Potential equivalence transformations for nonlinear diffusion-convection equations
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• Contact Equivalence Problem for Linear Parabolic Equations

Cited by in corpus (5)

• Point and generalized symmetries of the heat equation revisited
• Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• Lie reductions and exact solutions of dispersionless Nizhnik equation
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity