Construction of high order numerical methods for solving fourth order nonlinear boundary value problems

In this paper, we construct numerical methods of fourth, sixth and eighth orders convergence for solving fully fourth order nonlinear differential equation with the Dirichlet boundary conditions. The methods are based on the use of the trapezoidal quadrature formula with corrections for computing integrals at each iteration of the continuous iterative method for finding the solutions of the BVP. We get the error estimates for the actually obtained numerical solutions of the problem. Many numerical examples confirm the theoretical conclusions and show the efficiency of the proposed methods in comparison with some existing methods.