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.
@article{M2AN_2004__38_1_1_0, author = {Peradze, Jemal}, 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}, volume = {38}, year = {2004}, pages = {1-26}, doi = {10.1051/m2an:2004001}, mrnumber = {2073928}, zbl = {1080.35159}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 38 (2004) pp. 1-26. doi : 10.1051/m2an:2004001. http://gdmltest.u-ga.fr/item/M2AN_2004__38_1_1_0/
[1] On a class of functional partial differential equations. AN SSSR, Moscow, Selected Works. Izd. 3 (1961) 323-331.
,[2] Dynamic buckling of a nonlinear Timoshenko beam. SIAM J. Appl. Math. 34 (1979) 230-301. | Zbl 0423.73036
and ,[3] Théorie des vibrations. Béranger, Paris (1947). | JFM 65.1460.03
,[4] On an initial boundary value problem for the nonlinear Timoshenko beam. Ann. Acad. Bras. Cienc. 63 (1991) 115-125. | Zbl 0788.73038
,