A Comparative Numerical Study of Classical and Adaptive Methods for Solving Nonlinear Differential Equations Using MATLAB
DOI:
https://doi.org/10.65405/8xhtgm69Keywords:
nonlinear differential equations, numerical analysis, Runge–Kutta method, absolute error analysis, computational efficiencyAbstract
This study presents a comparative numerical analysis for solving a class of nonlinear differential equations for which closed-form analytical solutions are difficult to obtain. The aim of the study is to evaluate the efficiency of several numerical integration algorithms, including the Euler method, the fourth-order Runge–Kutta method (RK4), the implicit method, as well as the adaptive solver ode45 within the MATLAB environment, in terms of numerical accuracy, stability, computational time, and the effect of time step size. The methods were applied to three representative mathematical models exhibiting different types of nonlinear behavior. The absolute error, convergence rate, and sensitivity to time step variations were computed. The results showed that the RK4 method achieves the best balance between accuracy and computational efficiency, while ode45 provides the highest numerical stability in cases involving rapid variations. In contrast, the Euler method remains the least accurate despite its computational speed. The scientific contribution of this study lies in providing a unified comparative framework that combines error analysis, numerical verification of convergence order, and time-step sensitivity analysis, thereby supporting the selection of the most appropriate algorithm for scientific and engineering applications.
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