Hybrid Block Method for Numerical Solution of First Order Ordinary Differential Equations

This research introduces a novel single-step hybrid block method with four intra-step points that attains six-order accuracy, ensures A-stability, consistency, and convergence, and provides an efficient, accurate, and computationally economical tool for solving first-order ordinary differential equations. The formulation incorporates interpolation techniques to approximate function values at points where terms are not explicitly defined on the computational grid. In addition to the construction of the scheme, the paper rigorously investigates its theoretical properties. The results obtained show that the method not only achieves high accuracy but also performs competitively when compared with other established numerical techniques reported in the literature.