Improved 2-Point Hybrid Block Model for Direct Integration of Third and Fourth-Order Initial Value Problems

This research presents a novel numerical method for estimating solutions to higher-order differential equations with initial value problems (IVPs). By directly integrating a system of third and fourth-order differential equations using a 2-point implicit algorithm built with collocation and interpolation, significant improvements in efficiency are achieved. The proposed hybrid method, incorporating the first and second derivatives of the discrete algorithm, enhances accuracy and convergence rates. Convergence analysis confirms the zero-stability and consistency of the method, and numerical experiments demonstrate its superiority over existing methods for solving third and fourth-order equation systems. This work contributes a valuable numerical approach for efficient and accurate estimation of higher-order differential equations with IVPs.