Presentation of Simultaneous Ordinary Differential Equations 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
Simultaneous Ordinary Differential Equations
Two ordinary deferential equations
dy/dx=f(x,y,z) and dz/dx=g(x,y,z)
Initial condition (x_0,y_0,z_0)
Required y and z value at a given x value ?(x_n,y_n,z_n)
Step size ?_x=x_(i+1)-x_i
Number of steps n=(x_n-x_0)/?_x
¬The following steps are required:
y_(i+1)=y_i+1/6[k_1+?2k?_2+?2k?_3+k_4)
z_(i+1)=z_i+1/6[L_1+?2L?_2+?2L?_3+L_4)
where:
k_1=?_x×f(x_i,y_i,z_i )
L_1=?_x×g(x_i,y_(i,),z_i )
k_2=?_x×f(x_i+?_x/2,y_i+k_1/2,z_i+L_1/2)
L_2=?_x×g(x_i+?_x/2,y_i+k_1/2,? z?_i+L_1/2)
k_3=?_x×f(x_i+?_x/2,y_i+k_2/2,? z?_i+L_2/2)
L_3=?_x×g(x_i+?_x/2,y_i+k_2/2,? z?_i+L_2/2)
k_4=?_x×f(x_i+?_x ,y_i+k_3 ,? z?_i+L_3 )
L_4=?_x×g(x_i+?_x ,y_i+k_3 ,? z?_i+L_3 )