Abstract
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.