Optimized Hybrid Block Methods with High Efficiency for the Solution of First-order Ordinary Differential Equations

This article presents optimized hybrid block methods for solving first-order ordinary differential equations. The derivation employed the interpolation and collocation techniques using a three-parameter approximation. The hybrid points were obtained by minimizing the local truncation error of the main method. The obtained schemes were reformulated to reduce the number of function evaluations. The discrete schemes were derived as a by-product of the continuous scheme and used simultaneously to solve first-order initial value problems (IVPs). The schemes are self-starting, consistent, zero-stable, and convergent. The numerical results were compared with some existing techniques and found to be more accurate and efficient.