Maximizing Accuracy: Advancements in Numerical Methods for Ordinary Differential Equations

Abstract

Euler’s Method, Taylor’s Method are the most fundamental and easiest methods to solve first order ordinary differential equations (ODEs). Many other methods like Runge-Kutta Method have been developed on the basis of these method. In this paper, the basic ideas behind Euler's Method, Taylor's Method, and Runge-Kutta Method, as well as the geometrical interpretation have been discussed. The main focus is confined to the mathematical interpretation and graphical representation of these method and to find a way to reduce the errors. In order to verify the accuracy of these methods, we compare numerical solutions to exact solutions. Numerical experiment and graphical representation of a specific problem have been discussed in this paper. MATLAB programs have been used for graphical representation and FORTRAN programs have been used for computational efficiency.