A comparative study of iterative methods for solving nonlinear boundary value problems

Abstract

Nonlinear boundary value problems (BVPs) frequently arise in applied mathematics, physics, and engineering disciplines. Unlike linear BVPs, their nonlinear nature makes analytical solutions rare and impractical in most cases. As a result, a range of numerical iterative methods has been developed to solve these problems with varying degrees of efficiency and accuracy. This article presents a comprehensive comparative study of several iterative techniques, including the shooting method, finite difference method (FDM) with Newton-Raphson iteration, collocation methods, and the Adomian Decomposition Method (ADM). Each method is evaluated in terms of convergence behavior, stability, computational efficiency, and applicability to stiff and strongly nonlinear systems. A range of test cases from classical nonlinear physics and engineering models are examined, such as the Bratu problem, nonlinear reaction-diffusion equations, and thermal boundary layer models. Results indicate that while Newton-based FDM approaches demonstrate faster convergence for smooth problems, methods like ADM and collocation offer better performance for strongly nonlinear and singular problems. The article concludes by identifying strengths, weaknesses, and prospective improvements for each method, emphasizing the growing relevance of hybrid and AI-integrated iterative techniques in handling high-dimensional nonlinear systems.