A New Method for the Direct Solution of Second, Third and Fourth Order Differential Equations with Oscillatory Behavior

This paper presents a new block hybrid method for the direct numerical solution of second, third, and fourth-order ordinary differential equations (ODEs) with oscillatory characteristics. Unlike traditional approaches that rely on transforming higher-order equations into systems of first-order equations, the proposed method tackles the equations in their original forms, thereby reducing computational complexity and preserving the physical structure of the models. The method is formulated using power series approximations, collocation, and interpolation techniques to derive continuous implicit schemes and their discrete counterparts. A rigorous analysis of the method's basic properties such asrder, consistency, zero-stability, convergence, and absolute stability is conducted to establish its reliability. To validate the method, several numerical examples are presented. The results demonstrate that the new method offers high accuracy and stability compared to existing techniques, making it a robust and efficient tool for solving higher-order oscillatory differential equations directly.