Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

Abstract

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods.

Analysis of Reliable Solutions to the Boundary Value Problems by Using Shooting Method

“…Hofstrand et al [11] have discussed the bidirectional shooting method for extreme nonlinear optics. e authors have discussed some numerical methods for solving the initial value problems in [12][13][14]. Ahmad and Charan [15] make an attempt to compare the finite difference approach to the shooting method.…”