This study aims to construct an implicit block method with three-point to tackle general second-order ordinary differential equations (ODEs) directly. Hermite Interpolating Polynomial is used as the fundamental function to obtain the proposed method which involved the first and second derivatives of $f(t,u,u^{\prime })$. From the investigation done, it was found that the proposed method is consistent and zero-stable, hence it is convergent. The proposed method’s efficiency was obtained and a comparison was made in terms of accuracy to some existing methods with similar order or higher than it. This new method is able to solve linear and nonlinear initial value problems of the general second-order ODEs and outperformed existing methods with impressive results. Applications of the new method such as in the Fermi-Pasta-Ulam problem and van der Pol oscillator are discussed.

本研究旨在构建一种三点隐式块方法，以直接解决一般的二阶常微分方程（ODE）。以Hermite插值多项式为基本函数，以得到所提出的方法，该方法涉及的一阶和二阶导数。$F（Ť，ü，{ü}^{\prime }）$。从所做的调查中发现，所提出的方法是一致的并且是零稳定的，因此是收敛的。获得了所提出方法的效率，并在准确性方面与某些相似顺序或更高顺序的现有方法进行了比较。这种新方法能够解决一般二阶ODE的线性和非线性初值问题，并且性能优于现有方法，效果令人印象深刻。讨论了新方法在费米-帕斯塔-乌拉姆问题和范德波尔振荡器中的应用。