A Hybrid Fast Numerical Method for the Lane-Emden Differential Equation Using GHFs

Abstract

This paper introduces a novel hybrid numerical method which solves the Lane-Emden equation, leveraging Generalized Hat Functions (GHFs) of degrees 1 and 3 to achieve exceptional computational efficiency. By using linear GHFs for converting the equation into a block-structured nonlinear system solved via forward substitution, followed by cubic GHFs for refined approximation, the approach delivers up to 1000x speedup over direct cubic methods while maintaining L
∞ errors around 10−4. The proposed method adaptable to various nonlinear differential equations, it ensures 
consistent accuracy across interval lengths and extends seamlessly to fractional-order cases with minimal adjustments.