1. Numerical Calculations Part 5: Solving differential equations
Dr.Entesar Ganash

2. First order differential equation
The general first order differential equation is
Initial condition Is given
Example:

3. Rung-Kutta fourth-order formulae
There are a variety of alogorithms, under the general name of Runge-Kutta. This can be used to integrate systems of ordinary differential equations
In the first order differential equation:
First, Let us see this
The fourth- order Runge-Kutta estimate at is given by
where

4. Second order differential equation
The general second order differential equation is
Initial conditions are given
Example:

5. Rung-Kutta fourth-order formulae
In the second order, we have
where

6. Example
Write a program to solve the following differential equation in when use n=30
Solving
program RKM
implicit none
integer::n,i ! No of parts, counter
real ::y,dy,x,xn,h,exact
real ::k1,k2,k3,k4
n=30
x=0.0
y=0.0
dy=1.0
xn=1.0
h=(xn-x)/n
Exact=sin(x)
print*,' '
print*,' x', ' y', ' dy',' Exact'
print*, x,y,dy,Exact

7. continue Solving DO I=1,n k1=h*f (x,y,dy)
k2=h*f(x+h/2,y+h*dy/2+h*k1/8,dy+k1/2)
k3=h*f(x+h/2,y+h*dy/2+h*k1/8,dy+k2/2)
k4=h*f(x+h, y+h*dy+h*k3/2, dy+k3)
x=x+h
Exact=sin(x)
y=y+h*(dy+(k1+k2+k3)/6)
dy=dy+(k1+2.0*k2+2.0*k3+k4)/6
print*, x,y,dy,exact
End do
print*,' '
CONTAINS
REAL Function f(p,q,u)
Real ::f, p,q,u
f=-q
End function f
End program RKM
continue Solving

8. The result:

9. References
Hahn, B.D., 1994, Fortran 90 For Scientists and Engineers, Elsevier