Optimization of Two Hybrid Points in a One Step Method

Hybrid block methods have recently gained prominence in the numerical solution of first-order differential equations due to their enhanced stability and accuracy properties. This paper introduces an optimized one-step method that incorporates two hybrid points to improve computational efficiency and precision. By applying interpolation and collocation techniques, the block method is constructed by using a variable placeholder in place of a conventional fractional hybrid point, providing flexibility in optimizing the scheme. The optimal values of the hybrid points are obtained by equating the main method of the block to zero and solving the resulting equation. The derived method is of order four and is rigorously analyzed to be consistent, zero-stable, convergent, and A-stable, ensuring both reliability and long-term accuracy in practical computations. Results demonstrate that the proposed scheme consistently outperforms existing methods in the literature, including some of higher order, particularly in terms of accuracy and computational efficiency