Development and Analysis of a Modified Homotopy Analysis Method for Solving Non-Linear Fractional Order Lane-Emden Type Differential Equations

Abstract

This research presents a novel modification of the Homotopy Analysis Method (HAM) for solving non-linear fractional-order Lane-Emden type differential equations, which play a crucial role in various physical phenomena, particularly in astrophysics and mathematical physics. The proposed method introduces an adaptive convergence control parameter that dynamically adjusts based on the solution behavior, significantly enhancing the method's efficiency and convergence properties. Unlike traditional HAM approaches, our modification incorporates a specialized auxiliary function selection mechanism that better handles the singular nature of Lane-Emden equations while maintaining the flexibility of the fractional derivative definition. The mathematical framework is rigorously developed, and comprehensive convergence analysis is provided to establish the method's theoretical foundations. Through extensive numerical experiments on various test problems, we demonstrate that the modified method achieves superior accuracy compared to conventional techniques, including the standard HAM and Adomian decomposition method. Error analysis reveals that our approach reduces computational complexity while maintaining stability across different fractional orders. The results indicate particular effectiveness in handling strongly non-linear cases and singular behaviors characteristic of astrophysical applications. This development represents a significant advancement in the numerical treatment of fractional-order differential equations, offering improved reliability for practical applications in physics and engineering.