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, Tome 38 (2004) no. 1 , p. 1-26
Zbl 1080.35159 | MR 2073928
doi : 10.1051/m2an:2004001
URL stable : http://www.numdam.org/item?id=M2AN_2004__38_1_1_0

Classification:  35Q,  65M
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.

Bibliographie

[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