A Computationally Efficient Iterative Scheme for Solving Fourth-Order Boundary Value Problems

Abstract

A new rapid-converging analytical scheme is introduced to obtain the approximate analytical solutions of nonlinear fourth-order two-point boundary value problems, which appear in various physical phenomena. The idea of the method to obtain the solution of such problems is essentially based on reducing the solution of the main problem to the solution of an integral problem. The introduced technique consists of two steps. First, construct an integral operator by introducing Green’s function, and then, the Normal-S iterative scheme is applied to this integral operator to construct the iterative approach for such problems, which yields a simple way to improve the convergence of the iterative solutions to the problem. We also discuss the convergence of the introduced iterative method. To exhibit the performance of the method, we consider some numerical test examples. The obtained results are compared with the existing analytical and numerical approaches to reveal the superiority and computational efficiency of the proposed approach. In fact, it is a direct recursive and computationally cost-effective method for dealing with strong nonlinearity. The numerical simulations signify the applicability and effectiveness of the present work.