The existence of a solution and a numerical method for the Timoshenko nonlinear wave system
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 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 : 10.1051/m2an:2004001
Classification : 35Q, 65M
Mots-clés : Timoshenko nonlinear system, beam, Galerkin method, Crank-Nicholson scheme, Picard process 
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     title = {The existence of a solution and a numerical method for the {Timoshenko} nonlinear wave system},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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     publisher = {EDP-Sciences},
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     year = {2004},
     doi = {10.1051/m2an:2004001},
     mrnumber = {2073928},
     zbl = {1080.35159},
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Peradze, Jemal. The existence of a solution and a numerical method for the Timoshenko nonlinear wave system. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 1-26. doi : 10.1051/m2an:2004001. http://archive.numdam.org/articles/10.1051/m2an:2004001/

[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

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

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

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