Reduction operators of the linear rod equation

For linear partial differential equations, there exist well-developed classical methods of their analytical solution, which, in particular, includes the separation of variables, different integral transforms, Fourier series and their generalizations. At the same time, the study of symmetry properties of such equations is important, first of all, for the development of methods of symmetry analysis itself. In this paper we consider the (1+1)-dimensional constant-coefficient linear rod equation utt + λuxxxx = 0, where λ > 0, for unknown function u of the two independent variables t and x. This equation describes transverse vibrations of elastic rods. It is a special case of the Euler–Bernoulli beam equations, corresponding to constant values of parameters. Lie symmetries and the general equivalence problem for the class of Euler–Bernoulli beam equations were studied in [5,6,11]. By simple scaling of t or x, without loss of generality we can set λ = 1, i.e., it is sufficient to consider the equation