On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates

  • Jemal PeradzeEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)


The boundary value problem for a Timoshenko system of differential equations describing the static behavior of a plate is considered. Two sought functions are expressed through the third one for which an integro-differential equation with the Dirichlet condition on the boundary is written. The application of the Galerkin method to the obtained problem leads to a nonlinear system of algebraic equations which is solved by iteration. The condition of iteration process convergence is established and its rate is estimated.


Timoshenko equation Galerkin method Jacobi iteration Convergence of the iteration method 


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  1. 1.
    Birkhoff, G., Mac Lane, S.: A brief Survey of Modern Algebra. Second edn. The Macmillan Co., Collier-Macmillan Ltd., New York, London (1965)Google Scholar
  2. 2.
    Lagnese, J., Lions, J.-L.: Modelling analysis and control of thin plates. Recherches en Mathematiques Appliquees [Research in Applied Mathematics], 6. Masson, Paris (1988)Google Scholar
  3. 3.
    Lebedev L., P., Vorovich I., I.: Functional Analysis in Mechanics. Springer Monographs in Mathematics. Second edn. Springer, New York (2003)Google Scholar
  4. 4.
    Marchuk, G.I.: Methods of Numerical Mathematics. Transl. from the Russian by Arthur A. Brown. Applications of Mathematics, 2. Second edn. Springer, New York–Berlin (1982)Google Scholar
  5. 5.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Reprint of the 1970 original. Classics in Applied Mathematics, 30. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000)Google Scholar
  6. 6.
    Vol’mir, A. S.: Nonlinear Dynamics of Plates and Shells (in Russian). Second edn. Nauka, Moscow (1972)Google Scholar
  7. 7.
    Vorovich I., I.: Nonlinear Theory of Shallow Shells. Applied Mathematical Sciences, 133. Second edn. Springer, New York (1999)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Tbilisi State UniversityTbilisiGeorgia
  2. 2.Georgian Technical UniversityTbilisiGeorgia

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