Symmetry and new solutions of the equation for vibrations of an elastic beam (2009)
In this article we study the "no Lie" symmetry of the beam equation, all linear differential symmetry operators are constructed, up to the third order. It is found that the problem of solving this equation is reduced to the search for solutions of two Kolmogorov equations. Several kinds of solutions are cleared from the equation, particularly those that verify the initial areolar velocity conservation law and those that verify the initial elasticity conservation law. The equivalence between the Cauchy problem and the existence of a specific symmetry is illustrated. The striking parallelism that exists between the beam equation and the wave equation is found. Applying the "Ansatz method" builds a wide family of new exact solutions that particularly include those that describe the propagation of waves with damping. All the results of the article are new, the few known results in the literature are always mentioned.