Numerical Approximation Methods and Comparison with RK-4 Method for a Linear Differential Equation with Initial Conditions Using Scilab 6.1.1

Numerical approximation methods have been developed to linear differential equation with initial condition. The comparison of the results has been done among Euler’s method, modified Euler’s method and RK-4 with the help of Scilab software 6.1.1. RK-4 method is effective enough to reach more accuracy in the result. The Runge-Kutta method attempts to overcome the problem of the Euler's method, and modified Euler's method and the study shows that in all cases RK-4 method improves to a great extent, than those by the Euler method and Modified Euler’s Method. For RK-4 the approximation accuracy is proportional to the fourth power of the step size, thus making it a powerful and widely used numerical method also this method gives us higher accuracy without performing more calculations. Three different values of the step size have been taken. It is observed that smaller values of step size give better result; in all cases RK-4 method fits best as compared to others. The nature of the plot obtained by directly matches with the approximation solutions. It is observed that RK-4 method is suitable for obtaining the accurate solution of ODEs when the taken step sizes are too much small; since smaller h reduces the error.