Iterative Solution of a Nonlinear Static Beam Equation

Abstract

We consider a boundary-value problem for the nonlinear integrodifferential equation $$ {u}^{\prime \prime \prime \prime }-m\left(\underset{0}{\overset{l}{\int }}{u}^{\prime 2} dx\right){u}^{{\prime\prime} }=f\left(x,u,{u}^{\prime}\right),\kern1em m(z)\ge \upalpha >0,\kern1em 0\le z<\infty, $$ simulating the static state of the Kirchhoff beam. The problem is reduced to a nonlinear integral equation, which is solved by using the Picard iterative method. The convergence of the iterative process is established and the error is estimated.