Application of Homotopy Analysis Method on Selected Highly Nonlinear BVPs

This research explores the application of the Homotopy Analysis Method (HAM) to address selected highly nonlinear boundary value problems (BVPs) commonly found in physical and engineering sciences. Traditional approaches such as the perturbation method, homotopy perturbation method (HPM), and other semi-analytical techniques often face limitations due to the requirement for small parameters or lack of control over convergence. In contrast, HAM provides a flexible framework that introduces an auxiliary parameter, enabling convergence control of the solution series without relying on the existence of small parameters. The study is structured into three core chapters. The first chapter lays a comprehensive foundation, introducing key fluid dynamics concepts, heat transfer principles, types of differential equations, and mathematical laws pertinent to the subsequent analyses. Chapter two investigates the nonlinear convection-radiation heat transfer equation, applying HAM and comparing its effectiveness with the perturbation method and HPM. The analysis reveals that HAM maintains high accuracy even for large parameter values, where perturbative techniques fail due to asymptotic divergence. Using Mathematica, the convergence behavior is examined, and error profiles are plotted to validate the results. Section three presents a novel application of HAM to solve second- and fourth-order Sturm–Liouville eigenvalue problems, which are critical in modeling vibrations, thermal analysis, and elastic stability. The study introduces new algorithmic formulations and solution profiles, capturing multiple eigenvalue solutions and validating them through the appearance of λ-plateaus. These results showcase HAM’s capacity to yield multiple accurate eigenfunctions from a single initial approximation, highlighting its robustness and broader applicability compared to traditional methods. The outcomes confirm that the Homotopy Analysis Method is a powerful and adaptable tool for solving com