The existence of a solution and a numerical method for the Timoshenko nonlinear wave system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, p. 1-26

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank-Nicholson type scheme. The system of equations obtained by discretization is solved by a version of the Picard iteration method. The accuracy of the proposed algorithm is investigated.

DOI : https://doi.org/10.1051/m2an:2004001
Classification:  35Q,  65M
Keywords: Timoshenko nonlinear system, beam, Galerkin method, Crank-Nicholson scheme, Picard process
@article{M2AN_2004__38_1_1_0,
title = {The existence of a solution and a numerical method for the Timoshenko nonlinear wave system},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
pages = {1-26},
doi = {10.1051/m2an:2004001},
zbl = {1080.35159},
mrnumber = {2073928},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_1_1_0}
}

Peradze, Jemal. The existence of a solution and a numerical method for the Timoshenko nonlinear wave system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, pp. 1-26. doi : 10.1051/m2an:2004001. http://www.numdam.org/item/M2AN_2004__38_1_1_0/

[1] S. Bernstein, On a class of functional partial differential equations. AN SSSR, Moscow, Selected Works. Izd. 3 (1961) 323-331.

[2] M. Hirschhorn and E. Reiss, Dynamic buckling of a nonlinear Timoshenko beam. SIAM J. Appl. Math. 34 (1979) 230-301. | Zbl 0423.73036

[3] S. Timoshenko, Théorie des vibrations. Béranger, Paris (1947). | JFM 65.1460.03

[4] M. Tucsnak, On an initial boundary value problem for the nonlinear Timoshenko beam. Ann. Acad. Bras. Cienc. 63 (1991) 115-125. | Zbl 0788.73038