Iterative solution of a nonlinear static beam equation

TL;DR

This paper presents an iterative approach to solving a nonlinear boundary value problem modeling a Kirchhoff beam's static state, establishing convergence and error estimates for the method.

Contribution

It introduces a Picard iteration scheme for a nonlinear integro-differential equation of a Kirchhoff beam and proves its convergence with error bounds.

Findings

Convergence of the Picard iteration is rigorously established.

Error estimates for the iterative solution are derived.

The method effectively solves the nonlinear beam equation.

Abstract

The paper deals with a boundary value problem for the nonlinear integro-differential equation $u^{\prime\prime\prime\prime}-m\left(\int_0^l {u^\prime}^2dx\right)u^{\prime\prime}=f(x,u,u^\prime), \; m(z)\geq \alpha>0, \; 0\leq z <\infty$, modelling 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.