Iterative solution of a nonlinear static beam equation

G. Berikelashvili; A.  Papukashvili; J. Peradze

doi:10.37863/umzh.v72i8.833

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

DOI: https://doi.org/10.37863/umzh.v72i8.833

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.

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