Presentation of Runge-Kutta 4th Order Method I

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)







Numerical Solution of Ordinary Differential Equations

Undergraduate Level, 3th Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017

Runge-Kutta 4th Order Method
This method is utilized in solving ordinary deferential equation numerically. This method required an initial condition (x_0,y_0).

Ordinary Differential Equation
The ordinary deferential equation (ODE) dy/dx=f(x,y)
Initial condition (x_0,y_0)
Required y value at a given x value ?(x_n,y_n)
Step size ?_x=x_(i+1)-x_i
Number of steps n=(x_n-x_0)/?_x
¬The following steps are required:
f(x_(i+1),y_(i+1) )=y_(i+1)=y_i+1/6[k_1+?2k?_2+?2k?_3+k_4)
k_1=?_x×f(x_i,y_i )
k_2=?_x×f(x_i+?_x/2,y_i+k_1/2)
k_3=?_x×f(x_i+?_x/2,y_i+k_2/2)
k_4=?_x×f(x_i+?_x,y_i+k_3 )

Ex1: Find the value of the given ODE at f(0.2)
x+y-dy/dx=0
Initial condition: y(0)=1
Step size ?_x=0.1