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

- Volume: 38, Issue: 1, page 1-26
- ISSN: 0764-583X

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topPeradze, 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 38.1 (2004): 1-26. <http://eudml.org/doc/246009>.

@article{Peradze2004,

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.},

author = {Peradze, Jemal},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Timoshenko nonlinear system; beam; Galerkin method; Crank–Nicholson scheme; Picard process; existence; initial-boundary value problem for a beam; Crank-Nicholson type scheme; Picard iteration method},

language = {eng},

number = {1},

pages = {1-26},

publisher = {EDP-Sciences},

title = {The existence of a solution and a numerical method for the Timoshenko nonlinear wave system},

url = {http://eudml.org/doc/246009},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Peradze, Jemal

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 1

SP - 1

EP - 26

AB - 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.

LA - eng

KW - Timoshenko nonlinear system; beam; Galerkin method; Crank–Nicholson scheme; Picard process; existence; initial-boundary value problem for a beam; Crank-Nicholson type scheme; Picard iteration method

UR - http://eudml.org/doc/246009

ER -

## References

top- [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. Zbl0423.73036
- [3] S. Timoshenko, Théorie des vibrations. Béranger, Paris (1947). JFM65.1460.03
- [4] M. Tucsnak, On an initial boundary value problem for the nonlinear Timoshenko beam. Ann. Acad. Bras. Cienc. 63 (1991) 115–125. Zbl0788.73038

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