DIRECT COMPUTATIONAL APPROACHES FOR SOLVING SECOND-ORDER INITIAL VALUE PROBLEMS

This study presents the development and analysis of a new block hybrid method for the direct solution of second-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method is constructed to improve accuracy and stability in solving highly stiff and oscillatory systems that frequently arise in science and engineering applications. The theoretical analysis establishes the consistency, zero-stability, and convergence of the proposed scheme. To demonstrate its efficiency, the method is applied to a set of real-life problems and selected highly stiff test equations of the second order. Numerical results are compared with existing methods. The findings reveal that the new method provides smaller error magnitudes and better error control than its counterparts, particularly in stiff regimes where conventional techniques often fail. This confirms the robustness and reliability of the proposed method for solving second-order IVPs.