Abstract This numerical study exclusively focused on the direct two point diagonally multistep block method of order four (2DDM4) in the form of Adams-type formulas. The proposed predictor–corrector scheme was applied in this study to compute two equally spaced numerical solutions for the third-order two point and multipoint boundary value problems (BVPs) subject to Robin boundary conditions concurrently at each step. The optimization of the computational costs was taken into consideration by not resolving the equation into a set of first-order differential equations. Instead, its implementation involved the use of shooting technique, which included the Newton divided difference formula employed for the iterative part, for the estimation of the initial guess. Apart from studying the local truncation error, the study also included the method analysis, including the order, stability, and convergence. The results of eight numerical problems demonstrated and highlighted competitive computational cost attained by the scheme, as compared to the existing method.

中文翻译：

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三阶两点和多点Robin型边值问题的直接积分

摘要 本数值研究专门针对亚当斯型公式形式的四阶直接两点对角多步分块法 (2DDM4)。本研究中应用了所提出的预测器-校正器方案，以在每一步同时计算受罗宾边界条件约束的三阶两点和多点边界值问题 (BVP) 的两个等距数值解。通过不将方程分解为一组一阶微分方程来考虑计算成本的优化。相反，它的实现涉及使用射击技术，其中包括用于迭代部分的牛顿除法差分公式，用于估计初始猜测。除了研究局部截断误差，该研究还包括方法分析，包括阶次、稳定性和收敛性。与现有方法相比，八个数值问题的结果证明并突出了该方案获得的具有竞争力的计算成本。