Symmetry and new solutions of the equation for vibrations of an elastic beam

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.

• Autores: Sukhomlin, Nykolay; Alvarez, José R.
• Fecha de publicación: 2009
• Institución: Universidad EAFIT
• Repositorio: Repositorio EAFIT
• Idioma: spa
• OAI Identifier: oai:repository.eafit.edu.co:10784/14509
• Acceso en línea: http://hdl.handle.net/10784/14509
• Palabra clave: Operators Of Symmetry; Symmetry And Cauchy Problem; Parallelism Between Equations; Ansatz Method; Operadores De Simetría; Simetría Y Problema De Cauchy; Paralelismo Entre Ecuaciones; Método Ansatz
• License: Copyright (c) 2009 Nykolay Sukhomlin, José R. Alvarez