FORTRAN 90
Lecturer : Rafel Hekmat Hameed
University of Babylon
Subject : Fortran 90
College of Engineering
Year : Second B.Sc.
Mechanical Engineering Dep.
N
Nuum
meerriiccaall S
Soolluuttiioonn ooff FFiirrsstt O
Orrddeerr D
Diiffffeerreennttiiaall E
Eqquuaattiioonn
Euler Method
There are many different methods that can be used to approximate
solutions to a differential equation. We are going to look at one of the oldest
and easiest to use here. This method was originally devised by Euler and is
called, oddly enough, Euler’s Method. It is a simple method of solving firstorder ODE, particularly suitable for quick programming because of their great
simplicity, although their accuracy is not high.
Consider:
y' (x) = f(x, y)
;
y (x0) = y0
(1)
Let:
x i = x0 + i h ;
i = 0, 1,…. , n
yi = y (xi)
Yi
true solution evaluated at points xi
the solution to be calculated numerically.
Replace
y'(x)=(Yi+1-Yi)/h
Then Eq. (1) gets replaced with
Yi+1 = Yi + h f(xi , Yi)
EXAMPLE
Find a numerical solution to some first-order differential equation with
initial y(0) = 1, for 0  x  3.
5 dy/dx - y2 = -x2
dy/dx=1/5 (y2-x2)
The initial data is y(0) = 1 , assume h=.5 . So
x0 = 0 and y0 = 1.
x1 = x0 + h = 0 +.5=.5,
and
y1= y0 + h · f (x0, y0) = y0 +.1(y02 - x02)
˺
= 1 +.1(12 − 02)=1.1
x2 = x1 + h = .5 +.5=1
y2= y1 + h · f (x1, y1) = 1.1+.1 (1.12 − .52)=1.196
.
.
.
i
0
1
2
3
4
5
6
xi
0
.5
1
1.5
2
2.5
3
yi
1
1.1000
1.1960
1.2390
1.1676
.9039
0.3606
program Euler_method
implicit none
real, dimension (20)::x,y
integer::i
real :: h,xo,yo,f
read(*,*) xo,yo,h
x(1)=xo
y(1)=yo
print*,"for x=",x(1) , "y=",y(1)
i=2
do
x(i)=x(1)+(i-1)*h
y(i)=y(i-1)+h*f(x(i-1),y(i-1))
if (x(i).gt.3.0) exit
print*,"for x=",x(i) , "y=",y(i
i=i+1
enddo
end
real function f(x,y)
implicit none
real ::x,y
f=1/5.0*(y**2-x**2
;
end
˻