Preliminary group classification of equations v t t=f (x,v x)v x x+g(x,v x)

Abstract

A classification is given of equations vtt=f(v,vx)vxx+g(x,vx) admitting an extension by one of the principal Lie algebra of the equation under consideration. The paper is one of few applications of a new algebraic approach to the problem of group classification: the method of preliminary group classification. The result of the work is a wide class of equations summarized in Table II.

Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

“…It is remarkable that cases 4.7e and 4.7f are reduced exactly to equation (23). Exact solutions of equation (23) can be easily obtained by direct application of the classical Lie reduction method or using transformation (19) applied to solutions of equation (21).…”

“…Later, their investigation was generalized in [16,23,52] to equations of the following forms respectively…”

An Application of Equivalence Transformations to Reaction Diffusion Equations

“…We recall here that the principal Lie algebra L P [10,19] is the Lie algebra that leaves invariant the system (28) for any form of the functions f (u), Γ 1 (u), Γ 2 (v) and h(u, v). Then, the principal Lie algebra is the generator (29) where the functions α, β, δ and λ are solutions of the system (32)-(34) for arbitrary functions f (u), Γ 1 (u), Γ 2 (v) and h(u, v).…”

“…Following [11,15,16,19,20] (see also, e.g., [10,[21][22][23]), we look for the infinitesimal generator of the equivalence transformations of the system (1) of the form:…”

“…Following, e.g., [10], an equivalence transformation for the system (1) is a non-degenerate change of the independent and dependent variables t, x, u, v intot,x,û,v:…”