Numerical solution of the static beam problem by Bernoulli collocation method

We propose a numerical scheme to obtain an approximate solution of a boundary value problem for fourth order nonlinear integro-differential equation of Kirchhoff type. We first reduce the problem to a nonlinear finite dimensional system based on the Bernoulli polynomial approximation and then solve it by an iterative process together with a collocation method. Convergence of the iterative process and error estimates of the approximate solution are provided. Numerical experiments are conducted to illustrate the performance of the proposed method.