(untitled) (Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2020)
This paper presents an in-depth exploration of advanced numerical methods for solving ordinary differential equations (ODEs), essential for modeling and understanding complex physical systems. Traditional methods often fall short in terms of accuracy and efficiency when applied to non-linear or stiff ODEs, necessitating the development of more sophisticated techniques. This study focuses on several advanced methods, including Runge-Kutta methods, multistep methods, and finite element methods, detailing their theoretical foundations and practical applications. Comparative analyses are provided to highlight the strengths and limitations of each approach, supported by numerical experiments and error analysis. The implementation challenges and computational aspects are also discussed, offering insights into the choice of appropriate methods for different types of ODE problems. This work aims to serve as a comprehensive guide for researchers and practitioners in applied mathematics, engineering, and related fields, contributing to the advancement of numerical analysis and its applications in solving ODEs.