The preliminary group classification of the equation utt = f(x,ux)uxx + g(x, ux)
Narain, Ojen Kumar.
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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  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.