Numerical Methods for Solving Ordinary Differential Equations: A Comparative Study Using Engineering Problems

This study presents a comparative evaluation of widely used numerical methods for solving ordinary differential equations, with illustrative applications drawn from standard engineering problems relevant to Nigerian tertiary institutions. The methods under investigation include the Euler method, the improved Euler method, the fourth-order Runge-Kutta method, and the Adams-Bashforth predictor-corrector scheme. Problems involving population growth dynamics, heat transfer, and simple harmonic motion are formulated as initial value problems and solved using each method, with exact analytical solutions serving as benchmarks where derivable. Error analyses are conducted by computing absolute and relative errors at prescribed step sizes, and the computational efficiency of each method is assessed by recording iteration counts and convergence rates. The study is motivated by the growing reliance on numerical computation in engineering and applied science programmes at Nigerian universities, where students frequently lack exposure to systematic comparisons of approximation techniques. Programming implementations are developed in Python and MATLAB, and numerical outputs are tabulated and graphically presented for ease of interpretation. Results confirm that the fourth-order Runge-Kutta method consistently yields the lowest approximation error across all test problems, though at a higher per-step computational cost relative to simpler schemes. The Adams-Bashforth method demonstrates superior efficiency for problems requiring solution over extended intervals. Recommendations are made for incorporating comparative numerical experimentation into undergraduate mathematics curricula. Keywords: numerical methods, ordinary differential equations, Runge-Kutta, error analysis, undergraduate mathematics