This paper proposes an implicit block method with two-point to directly solve the fourth-order Initial Value Problems (IVPs). The implicit block method is derived by adopting Hermite interpolating polynomial as the basis function, incorporating the first derivative of to enhance the solution’s accuracy. A block formulation is presented to acquire the numerical approximation at two points simultaneously. The introduced method’s basic properties, including order, zero stability, and convergence, are presented. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed method compared with those of the several existing methods. Application in ship dynamics is also presented which yield impressive results for the proposed two-point block method.

中文翻译：

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直接用应用程序求解四阶初值问题的隐式两点分块方法

提出了一种含两点的隐式块方法来直接求解四阶初值问题（IVP）。隐式块方法是通过采用Hermite插值多项式作为基本函数，并结合了以提高解决方案的准确性。提出了一个块公式来同时获取两个点的数值逼近。介绍了所介绍方法的基本性质，包括阶数，零稳定性和收敛性。进行了数值实验，以验证该方法与几种现有方法的准确性和效率。还提出了在船舶动力学中的应用，对于所提出的两点分块方法，该方法产生了令人印象深刻的结果。