Application of Variable Step‐Size Hybrid Methods for Solving Third‐Order Lane‐Emden Equations

This manuscript introduces a pair of variable step‐size hybrid methods (PVSHM) to efficiently solve third‐order initial value problems of Lane‐Emden‐type equations (LETE). These equations are extensively used across various disciplines, including chemical engineering, fluid mechanics, physics, and astrophysics, to model a wide range of real‐world problems. The proposed method uses three intermediate points and a set of formulas specifically designed to avoid the singularity at the origin. In order to ensure optimal performance, the proposed PVSHM is implemented in variable step‐size mode using a suitable error estimation strategy that keeps the specified tolerances of the truncation errors under control. Numerical examples demonstrate the effectiveness of the proposed variable step‐size implementation, emphasizing the versatility of PVSHM in providing accurate and reliable results for applied third‐order IVP models in the form of LETE across diverse fields of study.