TAYLOR

 !Program name: TAYLOR

!Taylor method for differential equation dy/dx=-xy

!The exact solution: y=exp(-x^2/2)

!Interval X=0 to X=3 where y(0)=1

!y(n+1)= y(n)+h*(-xy)+(h^2)/2*[-y+(x^2)*y)

!Y(n+1) <=> X+h

 

program taylor

real :: x, y, h, exact, error

integer :: i

 

h=0.1

nstep=3./h ! 30 steps

y=1.   !y(0)=1

do i=0,nstep

   x=i*h   !It guarantees interval from 0 to 3

   y=y+h*(-x*y)+(0.5*(h**2))*(-y+(x**2)*y)

   exact=exp((-(x+h)**2)/2)

   error=exact-y

   print *, i, x+h, y, exact, error

!x+h is associated with Yn+1. Then x+h = Xn+1

!for h=0.1   y(1)=0.60526     error=0.00163

!for h=0.1   y(3)= 0.01256   error=-0.00015

end do

 

Print *,

h=0.05

nstep=3./h  !60 steps

y=1.   !y(0)=1

do i=0,nstep

   x=i*h    !It guarantees interval from 0 to 3

   y=y+h*(-x*y)+(0.5*(h**2))*(-y+(x**2)*y)

   exact=exp((-(x+h)**2)/2)

   error=exact-y

   print *, i, x+h, y,exact, error

!for h=0.05  y(1)=0.60621     error=0.000317

!for h=0.05  y(3)=0.011143    error=-0.000034

end do

stop

end