A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

中文翻译：

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求解u′′′=f(x,u(x))的二阶导数Runge-Kutta型方法应用于薄膜流动问题

提出并研究了一类涉及四阶导数的显式 Runge-Kutta 型方法，表示为二阶导数 Runge-Kutta 型 (TDRKT) 方法，用于求解形式为 u 的一类特殊的三阶常微分方程‴ ( x ) = f ( x , u ( x ) ) 。在本文中，提出了代数四阶的两个阶段和代数五阶的三个阶段。TDRKT 方法的推导涉及每一步的单三阶导数和四阶导数的多次求值。分析了方法的稳定性。通过数值实验展示了新方法的准确性和效率。