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Abstract

This paper introduces an innovative approach for solving fourth-order ordinary differential equations (ODEs) of the form. We present the embedded Runge-Kutta (RK) Direct Explicit (ERKDGF) method, a family of embedded direct explicit RK type methods tailored specifically for this purpose. Through meticulous application of Taylor expansion, we have derived algebraic equations with order conditions up to the sixth order, ensuring the accuracy and reliability of our proposed integrator. We have developed two key variants within this method, namely RKDF5(4) and ERKDGF5(4), with orders five and four, respectively. Our approach is strategically designed, with the higher-order method ensuring exceptional accuracy, and the lowerorder counterpart providing optimal error estimates. To facilitate practical implementation, we have devised a variable step-size code based on these methods, which was applied to solve a set of fourth-order problems. Our method’s performance was rigorously assessed through numerical experiments, with comparisons to existing embedded RK pairs that necessitate problem reduction into a system of first-order ODEs. The results unequivocally demonstrate the effciency of our ERKDGF method, both in terms of accuracy and the number of function evaluations required. This research marks a significant advancement in the field, offering a robust and effcient solution for directly solving fourth-order ordinary differential equations.

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