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The existence of a solution and a numerical method for the Timoshenko nonlinear wave system

Published online by Cambridge University Press:  15 February 2004

Jemal Peradze
Jemal Peradze*
Affiliation:
Department of Applied Mathematics and Computer Sciences of Tbilisi State University, Tbilisi, 380043, R. of Georgia. [email protected].
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Abstract

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.

Keywords

Timoshenko nonlinear systembeamGalerkin methodCrank–Nicholson schemePicard process.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 1 - 26
Copyright
© EDP Sciences, SMAI, 2004

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References

S. Bernstein, On a class of functional partial differential equations. AN SSSR, Moscow, Selected Works. Izd. 3 (1961) 323–331.
Hirschhorn, M. and Reiss, E., Dynamic buckling of a nonlinear Timoshenko beam. SIAM J. Appl. Math. 34 (1979) 230–301.
S. Timoshenko, Théorie des vibrations. Béranger, Paris (1947).
Tucsnak, M., On an initial boundary value problem for the nonlinear Timoshenko beam. Ann. Acad. Bras. Cienc. 63 (1991) 115–125.