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

Jemal Peradze

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Peradze, Jemal. "The existence of a solution and a numerical method for the Timoshenko nonlinear wave system." ESAIM: Mathematical Modelling and Numerical Analysis 38.1 (2010): 1-26. <http://eudml.org/doc/194206>.

@article{Peradze2010,
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},
keywords = {Timoshenko nonlinear system; beam; Galerkin method; Crank–Nicholson scheme; Picard process.; existence; initial-boundary value problem for a beam; Galerkin method; Crank-Nicholson type scheme; Picard iteration method},
language = {eng},
month = {3},
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/194206},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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; Galerkin method; Crank-Nicholson type scheme; Picard iteration method
UR - http://eudml.org/doc/194206
ER -

References

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  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).  Zbl65.1460.03
  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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