A comparative framework for convergence analysis of perturbation series techniques in nonlinear fractional quadratic differential equations

Abstract

This study tackles the challenge of obtaining highly accurate approximate solutions for nonlinear fractional differential equations, which often lack exact solutions due to their inherent complexity. A unified perturbation framework is proposed based on homotopy topology theory, enabling multiple formulations depending on the number of convergence-control parameters. Through dynamic adjustment of these parameters, the Homotopy Method achieves enhanced precision, particularly for fractional-order models exhibiting long-memory behavior. Numerical results clearly demonstrate that increasing the number of convergence parameters leads to significantly improved accuracy. Supported by detailed graphs and tables, the proposed approach proves to be a flexible, robust, and reliable tool for solving nonlinear fractional differential equations.