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

Abstract

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. At the first step, a kernel of the maximal algebras of invariance (MAIs) of the differential equations under study is found. It is proved that the one is a three-dimensional. A theorem about a “minimal” MAI of differential equations from the class is also formulated. At the second step, a group of equivalence transformations of the class under study is found. First, by using the infinitesimal method, the group of continuous equivalence transformations is calculated, which is then added to the complete equivalence group by two discrete transformations. At the third step, as a result of analysis of the system of determining equations, a theorem giving necessary conditions of the extension of the “minimal” MAI is formulated, namely, it is proved that a functional parameter involved in the class under study must satisfy one of the two Rikkati equations. Three examples of differential equations satisfying the necessary conditions of extension of the “minimal” MAI are considered. The MAIs of all equations are found. It is shown that among the examples considered the linear Kolmogorov equation admits the maximal symmetry properties.