a hybrid fast numerical method for the lane-emden differential equation using ghfs

This paper introduces a novel hybrid numerical method for solving the laneemden equation, leveraging generalized hat functions (ghfs) of different degrees 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. adaptable to various nonlinear differential equations, it ensures consistent accuracy across interval lengths and extends seamlessly to fractional-order cases with minimal adjustments.