## The preliminary group classification of the equation utt = f(x,ux)uxx + g(x, ux)

##### Abstract

We study the class of partial differential equations Utt = f(x, ux)uxx +
g(x, u x), with arbitrary functions f(x, u x) and g(x, u x), from the point of view
of group classification. The principal Lie algebra of infinitesimal symmetries
admitted by the whole class is three-dimensional. We use the method of preliminary
group classification to obtain a classification of these equations with
respect to a one-dimesional extension of the principal Lie algebra and then a
countable-dimensional subalgebra of their equivalence algebra. Each of these
equations admits an additional infinitesimal symmetry. L.V. Ovsiannikov [9]
has proposed an algorithm to construct efficiently the optimal system of an
arbitrary decomposable Lie algebra. We use this algorithm to construct an
optimal system of subalgebras of all dimensionalities (from one-dimensional
to six- dimensional) of a seven-dimensional solvable Lie algebra.