A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge Kutta Methods

Abstract

In this paper, we present Euler's method and fourth-order Runge Kutta Method (RK4) in solving initial value problems (IVP) in Ordinary Differential Equations (ODE). These two proposed methods are quite efficient and practically well suited for solving these problems. For us to obtain and verify the accuracy of the numerical outcomes, we compared the approximate solutions with the exact solution. We found out that there is good agreement between the exact and approximate solutions. We also compared the performance and the computational effort of the two methods. In addition, to achieve more accuracy in the solutions, the step size needs to be very small. Lastly, the error terms have been analyzed for these two methods for different steps sizes and compared also by appropriate examples to demonstrate the reliability and efficiency.