Development of a new iterative method and its convergence analysis for nonlinear fourth-order boundary value problems arising in beam analysis

In this study, an effective iterative technique based on Green's function is proposed to solve a nonlinear fourth-order boundary value problem (BVP) with nonlinear boundary conditions, which models an elastic beam. An iterative Green's function approach and a shooting method are integrated in the proposed method. The mathematical derivation is further supported by examining the convergence analysis of the proposed method. The iterative method rapidly generates a convergent series solution to the given problem. The proposed method has many advantages over existing methods: (a) A rapid convergence and (b) the ability to generate a highly accurate series solution with just a few iterations for complex problems arising in elastic beam. To demonstrate the applicability and effectiveness of the proposed method, four numerical examples are tested. The numerical results show that our methodology delivers more efficient results compared to other existing methods.