From Weak Nonlinear Perturbation to the Homotopy Analysis Method: A Rigorous Derivation and Theoretical Unification

Abstract

The Homotopy Analysis Method (HAM) is a widely used analytical approach for solving nonlinear problems, yet its theoretical foundation lacks rigorous justification, and its intrinsic correlation with perturbation theory remains ambiguous, leading to prevalent confusion in the existing literature. This study demonstrates that the fundamental homotopy deformation equation of HAM can be naturally derived from the weak-nonlinearity perturbation theory. We construct a specific analytical expression and optimize the core parameters (the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function) to mitigate the inherent strong nonlinearity of the nonlinear operator. Extending the small parameter \epsilon of perturbation theory to the interval [0,1] enables a systematic homotopy deformation process, which connects the linear auxiliary system (at \epsilon=0) with the original nonlinear problem (at \epsilon=1) and confirms HAM as a structured, adaptive generalization of classical perturbation theory. Furthermore, this work provides a rigorous proof that the Homotopy Perturbation Method (HPM) is a special case of HAM: HPM can be directly derived by fixing the optimal auxiliary linear operator as the linear component of the nonlinear system and setting the convergence-control parameter and auxiliary function to specific values, thus making HPM a degenerate form of HAM. This study clarifies the perturbation-theoretic origin of HAM, defines the hierarchical subordination of HPM to HAM, unifies the theoretical framework of homotopy-based nonlinear analytical methods, rectifies common misconceptions in the existing literature, and offers valuable guidance for the rational application, comparative analysis, and further development of such methods.