Example: Maximize flow for a trench section

The F90 code is written by

program app_1

implicit none

real :: x, up, upp, u, dx

real,parameter :: pi = 3.1416

integer :: i

u(x) = (1+cos(x))*sin(x)

! finite difference to find derivative

up(x) = (u(x+dx)-u(x-dx))/2./dx

upp(x) = (u(x+dx)+u(x-dx)-2.*u(x))/dx**2

dx = 1.*pi/180.

x = 45*pi/180.

do i= 1,10

x = x - 0.1*up(x)/upp(x)

print*,'i=',i,'x=',x*180./pi,'u(x)=',u(x)

end do

end program

The solution converges in about 10 iterations, where Theta to maximize flow area is 54.68 degrees giving flow area 1.287 m2.