A novel one-step optimized hybrid block method for solving general second-order ordinary differential equations

This study introduces a novel one-step hybrid block method for solving both stiff and non-stiff second-order ordinary differential equations (ODEs). The method incorporates optimization techniques and higher derivative functions, addressing the computational challenges of stiff ODEs while ensuring key numerical properties such as zero-stability, A-stability, and convergence. Numerical experiments demonstrate the method’s effectiveness across various problems, including classical examples from heat conduction, electrical circuits, and the Van der Pol oscillator. The results reveal that the proposed method achieves superior accuracy and efficiency, significantly outperforming existing methods in the literature. These findings underscore the potential of this approach as a robust and versatile tool for solving a wide variety of practical ODEs in engineering and applied sciences.