PAPER ON NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATION
DOI:
https://doi.org/10.29121/shodhkosh.v4.i2.2023.3461Keywords:
Ordinary Differential Equations, Numerical Methods, Galerkin’s Method, Runge-Kutta Methods, Stiff ODEs, Error Analysis, StabilityAbstract [English]
Ordinary Differential Equations (ODEs) are fundamental in modeling a wide range of phenomena in science, engineering, economics, and various other fields. However, finding exact analytical solutions to ODEs is not always possible, especially for complex, non-linear, or high-dimensional systems. In such cases, numerical methods offer a practical approach to approximating solutions. This paper explores various numerical techniques for solving ODEs, including the Euler method, Runge-Kutta methods, and multistep methods. It highlights the advantages and limitations of each approach in terms of accuracy, stability, and computational efficiency. The paper also discusses the concept of step size selection, error analysis, and the impact of discretization on the convergence of solutions. Additionally, we investigate specialized methods for stiff ODEs, which pose unique challenges due to the presence of rapidly changing variables alongside slow dynamics. Methods like the backward Euler and implicit Runge-Kutta schemes are reviewed for their effectiveness in handling stiffness. Finally, the paper presents several examples and applications to demonstrate the practical implementation of these methods and their ability to approximate solutions for real-world problems. We conclude by emphasizing the importance of choosing appropriate numerical techniques based on the nature of the problem at hand and the desired accuracy of the solution.
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