Numerical Solutions of Higher-Order Ordinary Differential Equations Using Runge–Kutta Methods with Extrapolation Techniques

Abstract

Higher-order ordinary differential equations (ODEs) play a central role in modeling complex dynamical systems, but their numerical solution remains challenging in terms of accuracy and computational efficiency. This study systematically investigates the performance of classical numerical methods—Euler’s method, Heun’s method, fourth-order Runge–Kutta (RK4), fifth-order Runge–Kutta (RK5), and the Runge–Kutta–Fehlberg method (RK45)—for solving fourth- and fifth-order homogeneous and non-homogeneous ODEs. To enhance precision, we integrate Richardson’s and Aitken’s extrapolation techniques with these methods and assess their impact on accuracy and efficiency. The novelty of this work lies in providing a fixed-step, systematic comparison of classical Runge–Kutta methods combined with extrapolation, an area that has not been sufficiently explored for higher-order ODEs. Approximate solutions are tabulated and visualized, with absolute errors and CPU times analyzed to evaluate trade-offs between accuracy and efficiency. The results show that RK5 with Aitken’s extrapolation achieves the highest accuracy for homogeneous problems, while RK45 with Aitken’s extrapolation is most effective for non-homogeneous cases. These findings highlight the practical value of extrapolation techniques in refining classical methods for advanced ODE problems.