This article presents the development and analysis of an optimized two-step hybrid block method designed for the direct numerical integration of second-order Initial Value Problems (IVPs) of the form \(y'' = f(x, y, y')\). The proposed scheme utilizes Euler’s polynomials as basis functions for the approximation, employing interpolation and collocation techniques to derive a continuous linear multistep method. To enhance the accuracy and stability of the block method, an optimization strategy is implemented to determine the optimal placement of two off-step points within the two-step framework. Theoretical analysis confirms that the resulting block method is consistent, zero stable, and convergent, achieving an order of accuracy of seven. The method is implemented as a simultaneous integrator, eliminating the need for predictors or separate starting values. Numerical experiments on moderately and highly stiff problems demonstrate the superior performance of the proposed algorithm, comparisons with existing methods in the literature reveal a significant reduction in absolute error. The results establish the optimized hybrid block method as a robust, efficient, and highly accurate tool for solving complex second-order mathematical models in science and engineering.
Adeniran et al. (Wed,) studied this question.