Iterative solution of a nonlinear static beam equation

  • G. Berikelashvili Georg. Techn. Univ., Tbilisi; A. Razmadze Math. Inst., Tbilisi, Georgia
  • A. Papukashvili I. Javakhishvili Tbilisi State Univ.; I. Vekua Inst. Appl. Math., Tbilisi, Georgia
  • J. Peradze Georg. Techn. Univ., Tbilisi; I. Javakhishvili Tbilisi State Univ., Georgia

Abstract

UDC 519.6

The paper deals with a boundary-value problem for the nonlinear integro-differential equation $u''''-m \bigg(\int _0^l {u'}^2\,dx \bigg)u''  = f(x, u, u'),$ $m(z)\geq \alpha>0,$ $0\leq z < \infty,$ modeling the static state of the Kirchhoff beam.  The problem is reduced to a nonlinear integral equation, which is solved using the Picard iteration method.  The convergence of the iteration process is established and the error estimate is obtained.

References

C. Bernardi, M. I. M. Copetti, Finite element discretization of a thermoelastic beam, Archive ouverte HAL-UPMC, 29/05/2013, 23 p.

S. Fucik, A. Kufner, ˇ Nonlinear differential equations, Stud. Appl. Mech., 2. Elsevier Sci. Publ. Co., Amsterdam etc. 359 pp., ISBN: 0-444-99771-7 (1980).

G. Kirchhoff, Vorlesungen uber mathematische Physik ¨ , I. Mechanik, Teubner, Leipzig (1876).

T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type, Discrete Contin. Dyn. Syst., 694 – 703, ISBN: 978-1-60133-010-9; 1-60133-010-3 (2007).

J. Peradze, A numerical algorithm for a Kirchhoff-type nonlinear static beam, J. Appl. Math., 2009, Article ID 818269 (2009), 12 p. https://doi.org/10.1155/2009/818269 DOI: https://doi.org/10.1155/2009/818269

J. Peradze, On an iteration method of finding a solution of a nonlinear equilibrium problem for the Timoshenko plate, Z. Angew. Math. und Mech., 91, № 12, 993 – 1001 (2011). https://doi.org/10.1002/zamm.201100016} DOI: https://doi.org/10.1002/zamm.201100016

K. Rektorys, Variational methods in mathematics, science and engineering, Springer Science & Business Media, 571 pp. ISBN: 90-277-0488-0 (2012).

H. Temimi, A. R. Ansari, A. M. Siddiqui, An approximate solution for the static beam problem and nonlinear integro-differential equations, Comput. Math. Appl., 62, № 8, 3132 – 3139 (2011), https://doi.org/10.1016/j.camwa.2011.08.026 DOI: https://doi.org/10.1016/j.camwa.2011.08.026

S. Y. Tsai, Numerical computation for nonlinear beam problems, M. S. thesis, Nat. Sun Yat-Sen Univ., Kaohsiung, Taiwan (2005).

S. Woinowski-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17, 35 – 36 (1950)

Published
18.08.2020
How to Cite
Berikelashvili , G., A. Papukashvili, and J. Peradze. “Iterative Solution of a Nonlinear Static Beam Equation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 8, Aug. 2020, pp. 1024-33, doi:10.37863/umzh.v72i8.833.
Section
Research articles