Fortran Programming for order differential equation

[ra_forever8]

Consider the first order differential equation
dy/dy = f(t,y) = -16 t^3 y^2
with initial condition y(0)=1

Using second order Adams-Bashforth method, write a Fortran programming to generate an approximate solution to the problem.


please forgive me for not trying because I really don't how to do.
The only thing I know is exact solution :(4t^4 + 1)^-1 which I have done it by hand. Help me

 

[SteamKing]

We can't teach you how to program. It would be better for you to find someone at your school who knows programming who could work with you.

 

[ra_forever8]

Program adams
Implicit None
Real, allocatable :: y(:),t(:)
Real:: yo, tmax, h, k1,k2,y1
Integer:: NI, i
Real, external :: f

Print*, 'Enter yo,Tmax,NJ'
read*, yo,Tmax,NI
h= Tmax/NI
Print*, 'This gives stepsize h=',h
allocate (t(0:NI), y(0:NI))

t(0)=0
yo=1
k1=0
k2= -16*h**3
y1 = yo + h/2 *( k1+ k2)
do i=2, NI
t(i)= i*h
y(i)= y(i-1) + (h/2)*(3*f(t(i-1))- f(t(i)))
end do
end program adams

Real Function f(t,y)
Real t
Real y
f = - 16*t**3*y**2
Return
End (I manage to do the code but i got the one error and one warning.
Warning: Variable y1 has been given value but never been used.
Error:Function f has been called with too few arguments

 

[Mark44]

When you include code in your post, put a [code[/color]] tag at the beginning and a [/code[/color]] tag at the end. This preserves whatever indentation you have used, and makes your code easier to read. I have done this below.

Program adams
Implicit None
Real, allocatable :: y(:),t(:)
Real:: yo, tmax, h, k1,k2,y1
Integer:: NI, i
Real, external :: f

Print*,  'Enter yo,Tmax,NJ'
read*, yo,Tmax,NI
h= Tmax/NI
Print*, 'This gives stepsize h=',h
allocate (t(0:NI), y(0:NI))

t(0)=0
yo=1
   k1=0
   k2= -16*h**3
   y1 = yo + h/2 *( k1+ k2) !See comment 1, below[/color]
do i=2, NI
  t(i)= i*h
  y(i)= y(i-1) + (h/2)*(3*f(t(i-1))- f(t(i))) !See comment 2, below[/color] 
  end do
   end program adams
   
   Real Function f(t,y)
   Real t
   Real y
   f = - 16*t**3*y**2
   Return
   End

(I manage to do the code but i got the one error and one warning.
Warning: Variable y1 has been given value but never been used.
Error:Function f has been called with too few arguments

1. Instead of y1, you should probably be using y(1), the first element of your array. You should delete the declaration of y1 up at the top as well.
2. Your function f needs to be called with two arguments. You are calling it with one argument.
Here's what you have: f(t(i-1))- f(t(i)).
What you should probably have is this: f(t(i-1), y(i - 1))- f(t(i), y(i)).

 

[ra_forever8]

When i test my code by entering
yo=1
Tmax= 2 (same Tmax=Tend)
NJ=20 (same NJ=NI)
This gives the h stepsize =0.10000
but at the same time it display Run-time error showing
Error 112,Reference to undefined variable, array element or function result(/UNDEF)
F - in file adams.f95 at line 29 [+0089]
main - in file adams.f95 at line 22 [+0531]
IS MY CODE INCORRECT OR I DID NOT TEST THE CODE PROPERLY?

 

[Mark44]

I'm not positive about this, but I think the problem you're having is caused by declaring f as external. IOW, I would change this line --

Real external :: f

to this--
Real :: f

External implies to me that the definition for the function is in another file, which isn't the case in your code.

 

[ra_forever8]

hi, I deleted the external and used Real :: f and tested with same number 1,2,20
but still giving same error: displaying Run-time error showing
Error 112,Reference to undefined variable, array element or function result(/UNDEF)
F - in file adams.f95 at line 39 [+0089]
main - in file adams.f95 at line 31 [+0531]

 

[Mark44]

I didn't notice this before, but you're missing a proper end statement for your function. It should look like this:

  Real Function f(t,y)
     Real t
     Real y
     f = - 16*t**3*y**2
     Return
  End Function f[/color]

 

[ra_forever8]

Hi, Thanks for your reply
I changed to End Function f , but the problem still not solved

 

[ra_forever8]

Run-time error points errors at
F - at line 39: f = - 16*t**3*y**2
main - at line 31 : y(i)= y(i-1) + (h/2)*(3*f(t(i-1), y(i - 1))- f(t(i), y(i)))

 

[bigfooted]

in the absence of a decent debugger, the simplest thing you can do is to add a lot of write statements, outputting all the variables used during the calculation. For instance, write out this during the do loop, after initializing t(i):
i, t(i), t(i-1)

 

[ra_forever8]

I added this code after do loop
write(10,*) i, t(i),t(i-1) ! This is line 29
Still giving Run-time error but this time error at line 29.

Would you please tell where else do i need to add write statements

 

[ra_forever8]

Program adams
Implicit None
Real, allocatable :: y(:),t(:)
Real:: yo, tend, h, k1,k2
Integer:: NI, i
Real,external ::f

!Asking to enter the initial vaule of yo, final time and numbers of step
Print*, 'Enter yo,Tend,NI'
read*, yo,Tend,NI
h= Tend/NI
Print*, 'This gives stepsize h=',h
allocate (t(0:NI), y(0:NI))

!Initial Conditions
t(0)=0
yo = 1

!After using runge kutta method, found out k1 =0 and k2= -16h^3,
k1=0
k2= -16*h**3

!we know that y(n+1) =y(n) + h/2(k1+K2) at n=0
y(1) = yo + h/2 *( k1+ k2)

! Loop through the number of steps to calculate the following at each step
do i=2, NI
t(i)= i*h
write(10,*) i, t(i),t(i-1)
!Second order Adam bashforth for all n
y(i)= y(i-1) + (h/2)*(3*f(t(i-1), y(i - 1))- f(t(i), y(i)))
end do
end program adams

!declaring function
Real Function f(t,y)
Real:: t
Real:: y
f = - 16*t**3*y**2
Return
End Function f(This is my program so far.)

 

[Mark44]

Here's your do loop:

do i=2, NI 
  t(i)= i*h 
  write(10,*) i, t(i),t(i-1) 
  !Second order Adam bashforth for all n 
  y(i)= y(i-1) + (h/2)*(3*f(t(i-1), y(i - 1))- f(t(i), y(i))) 
end do

In the first iteration of your loop, when i = 2, this is what is happening:
t(2) gets set to 2 * .1
The write statement prints 2, .2, and some garbage value for t(1). You get a garbage value because t(1) has never been initialized.

y(2) gets set to y(1) + .1 * (3 * f(garbage[/color], y(1)) - f(.2, y(2)[/color])

In the second red part, you're passing y(2) as the argument, but that value is not known yet.
In your first attempt, you had this for your code:
y(i)= y(i-1) + (h/2)*(3*f(t(i-1))- f(t(i)))

I changed this to what you see above, because your function f needs to have two arguments, and in your first code, you have only one argument. You need to check your notes for the Adam-Bashforth algorithm to see what should go in the second call to your function f. I was only guessing as to what should go there, and my guess is probably wrong.

 

[ra_forever8]

I chaged the Adam-Bashforth algorithm
y(i) = y(i-1) + (h/2)*(3*f(t(i-1), y(i - 1))- f(t(i-2), y(i-2)))
and i added t(1) = h
also i changed yo to y(0) and its working now :) Thank you very much Mark44

 

[Mark44]

You're welcome!

 

[ra_forever8]

Now i want to plot this partial derivative ∂f/∂y = |-32 (4t^4 +1)^-1 t^3| against t using fortran to generate a file or something for the MATLAB to plot the graph. Does fortran plot the graph?
Help me please.

 

[Mark44]

Fortran the programming language doesn't have any graphics capabilities, but there are libraries that you can link your Fortran code to that can do plotting.

It would probably be easier for you to write Fortran code that wrote the data points to a file, and then use that file as input for Matlab to plot it.

 

[ra_forever8]

What do you mean by libraries sir?
Would you please demonstrate some Fortran code as an example for the Matlab to plot it.

 

[Mark44]

What do you mean by libraries sir?

A library is a collection of subroutines and functions that are compiled into object code, and that you can link your program to.

Would you please demonstrate some Fortran code as an example for the Matlab to plot it.

When the program you wrote is finished, there are two arrays that are completely initialized: the t array and the y array.

In addition to displaying this data in your program, you could use a write statement to store each pair of numbers (t, y) in a file. You could then use that file as input to a MATLAB script to graph the data.

 

[ra_forever8]

partial derivative ∂f/∂y = |-32 (4t^4 +1)^-1 t^3|
how to write this partial derivative ∂f/∂y in fortran.
my adam bashforth second order scheme interval of absolute stabilty: -1<λh<0
∂f/∂y= |λ|
so, h< 1/|λ|
h<1/ |-32 (4t^4 +1)^-1 t^3|
i need to plot ∂f/∂y against time
fortran code:
open(10,file='adams_results.m')
do i = 0
t(i) = i*h
∂f/∂y = |-32 (4t^4 +1)^-1 t^3|
∂f/∂y=|λ|
end do
write(10,*) "plot (time, |λ|)"
i got soo many errors

 

[Mark44]

partial derivative ∂f/∂y = |-32 (4t^4 +1)^-1 t^3|
how to write this partial derivative ∂f/∂y in fortran.
my adam bashforth second order scheme interval of absolute stabilty: -1<λh<0
∂f/∂y= |λ|
so, h< 1/|λ|
h<1/ |-32 (4t^4 +1)^-1 t^3|
i need to plot ∂f/∂y against time
fortran code:
open(10,file='adams_results.m')
do i = 0
t(i) = i*h
∂f/∂y = |-32 (4t^4 +1)^-1 t^3|
∂f/∂y=|λ|
end do
write(10,*) "plot (time, |λ|)"
i got soo many errors

This doesn't look much like Fortran code. You can't use ∂ in a variable name, nor can you use Greek letters such as λ.
You can't assign a value to an expression. IOW, you can't do this:
∂f/∂y = <some value>

You can't do this, either:
df/dy = <some value>

Your variable name has to start with a letter of the alphabet, and can't include symbols such as +, =, *, or /.

Your do loop is missing a required part: do i = 1, NI[/color]

I don't know what you're trying to do with this code --

∂f/∂y = |-32 (4t^4 +1)^-1 t^3|
∂f/∂y=|λ|

Aside from the already mentioned problems with illegal variable names, why are you assigning a value to df/dy, and then assigning a different value? That doesn't make any sense.

Fortran doesn't understand math symbols such as | | for absolute value. If you need the absolute value of some expression, use the abs() intrinsic function.


In your Fortran code here, you are using a variable, t, that hasn't been initialized. You are forgetting that you have an array named t, whose values are t(0), t(1), t(2), ..., t(NI). The array variable is what you should be using.

The biggest mistake is that you are writing brand-new code, when all you need to do is open a file for writing output, and then write the variables that you have already calculated in your previous version. To do this WRITE those t and y values into the file you opened.

 

[ra_forever8]

Thanks mark44, i will come back later for this question