# Symmetry Analysis of a Class of (2+1)-Dimensional Linear Ultra-Parabolic Equations

In this paper, a class of (2+1)-dimensional linear ultra-parabolic equations of the second order is investigated by using the methods of group analysis of differential equations. The class under study generalizes a number of the classical equations of mathematical physics such as the free Kramers equation, the linear Kolmogorov equation etc. The classification of the symmetry properties of equations from the class is carried out by using the well-known Lie–Ovsiannikov algorithm.

# Group Classification of Kolmogorov Nonlinear Equations

We consider Kolmogorov nonlinear equations with an arbitrary function. The group-theoretical method is one of the methods for solving partial differential problem. Using this method, we integrate equations with a non-trivial symmetry group. Therefore, group classification is high priority. Specifically, we conduct the group classification of Kolmogorov nonlinear equations. Using obtained continuous equivalence transformations, we present nonequivalent subclasses of these equations. We calculate maximum invariance algebras for all these subclasses.

# Preliminary Group Classification of a Class of Generalized Linear Kolmogorov Equations

The group–theoretic method is a modern research method for studying both linear and nonlinear partial differential equations. By using this method, we construct exact partial classical solutions of equations allowing for non-trivial symmetry groups. In this paper, a class of (2+1)-dimensional generalized linear Kolmogorov equations is considered. Our aim is to investigate symmetry properties of equations from the class and to use them to construct invariant fundamental solutions.

# Symmetry Properties and Exact Solutions of the Linear Kolmogorov Equation

This paper considers the symmetry properties of the linear Kolmogorov equation. We obtain the maximal invariance algebra of this equation. Moreover, we classify all two-dimensional subalgebras of the invariance algebra up to action of transformations of its automorphism group. Using the obtained subalgebras, we reduce the symmetry to ordinary differential equations and separate variables for this equation. In some cases we integrate the reduced equations and to obtain exact solutions of the linear Kolmogorov equation.