Fehlberg Runge-Kutta Method for Solving System of Ordinary Differential Equations

Abstract

Numerical methods are essential for solving linear and nonlinear ordinary differential equations (ODEs) encountered in various scientific and engineering applications. The objective of this study is to compare the fourth-order Runge-Kutta (RK4) and Fehlberg Runge-Kutta (RKF45) methods for solving such equations, focusing on their accuracy and efficiency under a fixed step size. The methods were implemented using MATLAB, enabling solution computations, graph plotting, and error analysis. The results show that RK4 achieves higher accuracy for smooth and predictable systems due to its simplicity and stability. In contrast, RKF45, while effective for complex systems, demonstrated reduced accuracy when constrained to a fixed step size. RKF45's potential is best realized when its adaptive step-sizing feature is employed. This study highlights the importance of selecting numerical methods based on the problem’s characteristics and computational goals. Both RK4 and RKF45 are reliable tools, but their performance depends on proper implementation and problem dynamics.