Symmetry and new solutions of the equation for vibrations of an elastic beam
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Keywords
operators of symmetry, Cauchy problem and symmetry, parallelism between equations, Ansatz method.
Abstract
In this short paper it is studied the “not Lie” symmetry of the beam equation. All operators of symmetry linear differential until third order are constructed. It is noted that the resolution of this equation reduces to finding solutions of two Kolmogorov equations. Several class of new solutions of the beam equation are found, particularly those that verify the law of conservation of the initial areolar speed and other that verify the law of conservation of the initial elasticity. It is showed the equivalence between the solution of the problem Cauchy and the existence of a specific symmetry. It is found a parallelism between the beam equation and the wave equation. Using the Ansatz method a wide new family of exact solutions is built. This family includes particularly the solutions which describe the propagation of damped waves. All results of this paper are new.
PACS: 43.40.+s
MSC: 83C05, 82B23
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References
[2] Maria Luz Gandarias and Maria de los Santos Bruzón. Classical and nonclassical symmetries of a generalized Boussinesq equation. Journal of Nonlinear Mathematical Physics, ISSN 1402–9251, 5(1), 8–12 (1998).
[3] Maria de los Santos Bruzón, José Carlos Camacho y José Ramírez Labrador. Modelo de vibraciones de una viga. Reducciones por simetrías, Proceeeding of the CISCI: International Conference on Education & Information Systems, Florida, USA, 168–193 (2004).
[4] Wafo Soh. Euler–Bernoulli beams from a symmetry standpoint–characterization of equivalent equations, http://arxiv.org/abs/0709.1151v1 (Preprint submitted to Elsevier). 7 September 2007.
[5] Rudolf Rothe. Matemática superior para matemáticos, físicos e ingenieros, ISBN 0–378–20663–1. Editorial Labor, S.A., Espa˜na, III, 1960. Referenciado en 28
[6] S. D. Conte. A stable implicit finite difference approximation to a fourth order parabolic equation. Journal of the Association for Computing Machinery, ISSN 0004–5411, 4(1), 18–23 (1957).
[7] V. N. Shapovalov. Group properties of linear equations. Russian Physics Journal, ISSN 1064–8887, 11(6), 75–80 (1968).
[8] N. Sukhomlin y M. Arias. Estudio de simetría y de posibilidades de resolución exacta de las ecuaciones de Sch¨odinger y Hamilton–Jacobi para un sistema aislado. Ciencia y Sociedad, ISSN 0378–7680, 29(1), 26–37 (2004).
[9] V. N. Shapovalov and N. B. Sukhomlin. Separation of variables in the non stationary Schr¨odinger equation. Russian Physics Journal, ISSN 1064–8887 (Print), 573–9228 (Online), 17(12), 1718–1722 (1974).
[10] N. Tijonov y A. A. Samarsky. Ecuaciones de la física matemática, ISBN 0–158– 70015–3. Mir, Moscú, 227 (1972).
[11] N. Sukhomlin. Conservation law of strike price and inversion of the Black- Scholes formula. Russian Physics Journal, ISSN 1064–8887 (Print) 1573–9228 (Online), 50(7), 741–743 (2007).
[12] H. Bethe. Zur theorie der metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Zeitschrift f¨ur Physik A Hadrons and Nuclei, ISSN 0939–7922 (Print) 1431–5831 (Online), 71(3–4), 205–226 (1931).
[13] Biao Li and Yong Chen. Nonlinear partial differential equations solved by projective Riccati equations Ansatz . Zeitschrift f¨ur Naturforschung A, ISSN 0932–0784, 58a, 511–519 (2003).
[14] Nikolay Sukhomlin and Jan Marcos Ortiz. Equivalence and new exact solutions to the Black–Scholes and diffusion equations. Applied Mathematics E-Notes, ISSN 1607–2510, 7, 206–213 (2007).