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Similarity reductions of a nonlinear model for vibrations of beams
| 1. | Departament of Mathematics, University of Cdiz, Cadiz, Spain, Spain |
References:
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M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, New York: Dover, (1972). Google Scholar |
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G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation,, {\em J. Math. Mech.}, 18 (1969), 1025. Google Scholar |
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J.M. Ball, Initial boundary value problems for an extensible beam,, {\em J. Math. Analysis Appl.}, 42 (1973), 61. Google Scholar |
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G. W. Bluman and S. Kumei, Symmetries and differential equations,, Springer-Verlag, (1989). Google Scholar |
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E. Burgreen, Free vibrations of a pinended column with constant distance between pinendes,, {\em J. Appl. Mech}, 18 (1951), 135. Google Scholar |
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B. Champagne, W. Hereman, and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations,, {\it Com. Phys. Comm.}, 66 (1991), 319. Google Scholar |
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R. W. Dickey, Free vibrations and dynamics buckling of extensible beam,, {\em J. Math. Analysis Appl.} {\bf 29} (1970)., 29 (1970). Google Scholar |
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P. A. Djondjorov, Invariant properties of timoshenko beam equations,, {\em International Journal of Engineering Science}, 33 (1995), 2103. Google Scholar |
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J. G. Eisley, Nonlinear vibrations of beams and rectangular plates,, {\em Z. angew. Math. Phys.}, 15 (1964). Google Scholar |
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J. Ferreira, R. Benabidallah, and J. E. Mu{\ n}oz Rivera, Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain,, {\em Rendiconti di Matematica, 19 (1999), 177. Google Scholar |
| [11] |
L. A. Medeiros, On a new class of nonlinear wave equations,, {\em J. Math. Appl.}, 69 (1979), 252. Google Scholar |
| [12] |
G. P. Menzala, On classical solutions of a quasilinear hyperbolic equation,, {\em Nonlinear Analysis}, 3 (1978), 613. Google Scholar |
| [13] |
D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation,, {\em Nonlinear Analysis}, 14 (1990), 613. Google Scholar |
| [14] |
O. C. Ramos, Regularity property for the nonlinear beam operator,, {\em An. Acad. Bras. Ci.}, 61 (1989), 15. Google Scholar |
| [15] |
J. E. M. Rivera, Smoothness effect and decay on a class of non linear evolution equation,, {\em Ann. Fac. Sc. Toulouse}, (1992), 237. Google Scholar |
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P. J. Olver, Applications of Lie groups to differential equations,, Springer-Verlag, (1986). Google Scholar |
| [17] |
S. K. Woinowsky, The effect of axial force on the vibration of hinged bars,, {\em Appl. Mech.}, 17 (1950), 35. Google Scholar |
show all references
References:
| [1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, New York: Dover, (1972). Google Scholar |
| [2] |
G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation,, {\em J. Math. Mech.}, 18 (1969), 1025. Google Scholar |
| [3] |
J.M. Ball, Initial boundary value problems for an extensible beam,, {\em J. Math. Analysis Appl.}, 42 (1973), 61. Google Scholar |
| [4] |
G. W. Bluman and S. Kumei, Symmetries and differential equations,, Springer-Verlag, (1989). Google Scholar |
| [5] |
E. Burgreen, Free vibrations of a pinended column with constant distance between pinendes,, {\em J. Appl. Mech}, 18 (1951), 135. Google Scholar |
| [6] |
B. Champagne, W. Hereman, and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations,, {\it Com. Phys. Comm.}, 66 (1991), 319. Google Scholar |
| [7] |
R. W. Dickey, Free vibrations and dynamics buckling of extensible beam,, {\em J. Math. Analysis Appl.} {\bf 29} (1970)., 29 (1970). Google Scholar |
| [8] |
P. A. Djondjorov, Invariant properties of timoshenko beam equations,, {\em International Journal of Engineering Science}, 33 (1995), 2103. Google Scholar |
| [9] |
J. G. Eisley, Nonlinear vibrations of beams and rectangular plates,, {\em Z. angew. Math. Phys.}, 15 (1964). Google Scholar |
| [10] |
J. Ferreira, R. Benabidallah, and J. E. Mu{\ n}oz Rivera, Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain,, {\em Rendiconti di Matematica, 19 (1999), 177. Google Scholar |
| [11] |
L. A. Medeiros, On a new class of nonlinear wave equations,, {\em J. Math. Appl.}, 69 (1979), 252. Google Scholar |
| [12] |
G. P. Menzala, On classical solutions of a quasilinear hyperbolic equation,, {\em Nonlinear Analysis}, 3 (1978), 613. Google Scholar |
| [13] |
D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation,, {\em Nonlinear Analysis}, 14 (1990), 613. Google Scholar |
| [14] |
O. C. Ramos, Regularity property for the nonlinear beam operator,, {\em An. Acad. Bras. Ci.}, 61 (1989), 15. Google Scholar |
| [15] |
J. E. M. Rivera, Smoothness effect and decay on a class of non linear evolution equation,, {\em Ann. Fac. Sc. Toulouse}, (1992), 237. Google Scholar |
| [16] |
P. J. Olver, Applications of Lie groups to differential equations,, Springer-Verlag, (1986). Google Scholar |
| [17] |
S. K. Woinowsky, The effect of axial force on the vibration of hinged bars,, {\em Appl. Mech.}, 17 (1950), 35. Google Scholar |
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