ABSTRACT

In this paper, an efficient fourth-derivative two-step hybrid block strategy (FDTHBS) to get the approximate solution of a third-order IVP with applications to problems in Fluid Dynamics and Engineering is constructed. The mathematical derivation of the proposed FDTHBS is based on the interpolation and collocation of the exact solution and its derivatives at the selected equidistant grid and off-grid points. The theoretical characteristics of the proposed method are analysed. An embedding-like procedure is considered and executed in variable step-size mode to get better performance of the newly developed strategy. Some test problems, including the well-known Blasius equation and three different types of non-linear thin-film flow problems, are integrated numerically to ascertain the superior impact of our developed error estimation and control strategy. It is worth concluding that the proposed technique is not only efficient in term of CPU time, but also minimizes errors and support the analytical results.

摘要

在本文中，构建了一种有效的四阶导数两步混合块策略（FDTHBS）来获得三阶IVP的近似解，并应用于流体动力学和工程问题。所提出的 FDTHBS 的数学推导基于精确解及其在选定等距网格和离网点处的导数的插值和搭配。分析了所提方法的理论特点。考虑并以可变步长模式执行类似嵌入的过程，以获得新开发策略的更好性能。一些测试问题，包括著名的 Blasius 方程和三种不同类型的非线性薄膜流动问题，数值积分以确定我们开发的误差估计和控制策略的卓越影响。值得得出的结论是，所提出的技术不仅在 CPU 时间方面是有效的，而且还可以最大限度地减少错误并支持分析结果。