A Variational-Fixed Point Iteration Method with Rayleigh-Ritz Initialization for Two-Point Boundary Value Problems

Abstract

Boundary value problems often require an initial approximation that is sufficiently close to the desired solution for an iterative procedure to converge efficiently. In this study, a Variational-Fixed Point Iteration Method with Rayleigh–Ritz Initialization (VFPIM-RR) is developed for solving second-order two-point boundary value problems. The main idea is to use the Rayleigh–Ritz method to construct a boundary-compatible initial approximation before applying variational and fixed-point corrections. Unlike approaches that begin with an arbitrarily selected or simple initial function, the proposed procedure obtains the starting approximation from the variational structure of the problem and subsequently improves it through a Variational Iteration Method correction and a Green's-function-based fixed-point iteration. A Lagrange multiplier is derived for the correction functional, while sufficient conditions for convergence of the resulting hybrid operator are established using the Banach Fixed-Point Theorem. The effectiveness of the method is examined using two benchmark boundary value problems with known exact solutions. The numerical results show that the Rayleigh–Ritz initialization provides a useful starting approximation and that successive hybrid iterations rapidly reduce both the approximation error and the governing-equation residual. In the reported tests, the method attains errors of approximately after only a few iterations. The results demonstrate that coupling a variationally optimized initial approximation with residual correction and fixed-point refinement provides an accurate and computationally effective framework for second-order two-point boundary value problems.