Analytical and Numerical Study of Runge-Kutta Methods for the Approximate Solution of Ordinary Differential Equations

Abstract

This research presents a comprehensive analytical and numerical investigation of Runge–Kutta methods for the approximate solution of initial value problems (IVPs) arising in ordinary differential equations. The study focuses on three variants: the second-order Runge Kutta method (Heun’s method), the classical fourth-order Runge Kutta method, and the fifth-order Runge–Kutta method (Butcher’s method). Each method is systematically implemented using MATLAB, and their numerical solutions are compared against exact analytical results to assess accuracy and stability. A series of computational experiments with varying step sizes (0.1, 0.05, and 0.025) are performed, with outcomes presented in tabular form to illustrate convergence trends and error propagation. The analysis of error terms, approximate values, and maximum errors enables a rigorous comparative evaluation of these schemes. The findings reveal that although all methods benefit from reduced step lengths, higher-order Runge-Kutta algorithms, particularly the fifth-order formulation, demonstrate superior accuracy and computational efficiency. This work highlights the essential role of Rung- Kutta methods in the robust numerical approximation of differential equations, with direct implications for applied mathematical modelling and scientific computing.