# Homework Help: Solving PDE heat problem with FFCT

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1. Sep 3, 2017

### Orodruin

Staff Emeritus
What exactly do you mean by "substituting the $U_1$"? Please write it out explicitly, it helps for understanding your thought process and whether you understood previous replies or not.

2. Sep 3, 2017

### Aows

t
this is my last step:

$$\frac {dU} {dt} = - \left( \frac {{n} {\pi}} L \right)ˆ(2) * U(x,t) + \left( {-1} \right)ˆn * U_1$$

3. Sep 3, 2017

### Aows

after my last step, i should separate the variables but if i have $U_1$, i don't know hot to perform the separation with the $U_1$ exist .... @Orodruin

4. Sep 3, 2017

### Orodruin

Staff Emeritus
You should get rid of the inhomogeneous boundary condition before you attempt the transform. Essentially you can do this by the ansatz $u(x,t) = v(x,t) + h(x)$ where $h(x)$ is the stationary solution for your inhomogeneous boundary conditions. You will then get an ODE for $v(x,t)$ that you can solve using either the eigenfunctions proposed in #13 or by the extension proposed in #19.

So first step: What is the stationary solution?

If you have problems with this, you can also start by solving the problem for $U_1=0$ and deal with this in the end. In that case, do you understand the odd extension around $x=L$?

5. Sep 3, 2017

### Aows

Mr. @Orodruin , I really didn't understand this, why not write the answer after my last step ?

6. Sep 3, 2017

### Aows

I didn't use this anstaz before $u(x,t) = v(x,t) + h(x)$ @Orodruin

7. Sep 4, 2017

### Aows

Hello Mr. @Orodruin , any updates regarding the problem ?

very thankful,
Aows K.

8. Sep 5, 2017

### RUber

You should try to do this.
At time = 0, you get
$u(x,0)=v(x,0) + h(x)$
Your cosine transform is just in terms of x, right? And for each n, you have an ODE to solve in terms of a function of t, which should be of the form:
$f'(t) = g(t) + c$
where your functions of x are treated as constants in terms of t. What methods do you know for solving ODEs of this type?

If you use separation of variables, the function might look like:
$u(x,t) = v(x)f(t) + h(x)$
or in the transformed space
$U_n(x,t) = a_n \cos (n \omega x) f(t) + b_n \cos (n \omega x) .$

9. Sep 5, 2017

### Orodruin

Staff Emeritus
I strongly suggest you follow my approach. A priori, you are only allowed to expand in the cosines if the function satisfies the appropriate homogeneous boundary conditions. Also, there cannot be any $x$ dependence left after the Fourier transform - you are transforming it away, it is the entire point, to get an $x$ independent differential equation for every Fourier mode!

10. Sep 6, 2017

### Aows

I don't know how to use your approach, can you help with the rest of the solution ?

11. Sep 6, 2017

### Orodruin

Staff Emeritus
I have already pointed out several mistakes you have made and told you that your approach will not work and why. I cannot help you more unless you specify exactly what it is that you are having trouble with in the proposed approach.

12. Sep 6, 2017

### Aows

i have another example it is also using FFCT but the value of the first derivative is zero which makes it easy to solve, here is this example the value of the first derivative is #U_1# which i dont know how to deal with it, this is my problem...
add to that, i dont know how to use your approach, we never use it before ...
your help is highly appreciated,... @Orodruin

13. Sep 6, 2017

### Orodruin

Staff Emeritus
But I described to you how to use it. What in that description poses a problem?

14. Sep 9, 2017

### Aows

I don't know how to apply it, because i didn't use it before... @Orodruin

15. Sep 14, 2017

### Aows

Hello dear gents, @RUber @Orodruin ,
my exam will take place on saturday, can you provide a full detailed answer for the problem or not ?

regards,
Aows K.

16. Sep 14, 2017

### Orodruin

Staff Emeritus
What you are asking is agains the forum rules. You need to solve the problem yourself based on the hints that you have been given. If there are things you do not understand about those hints, ask about it specifically.

17. Sep 14, 2017

### Aows

dear Mr. @Orodruin ,
i read the forum rules since the first day that i signed up, and it says that you need to show your attempt so others can help with what you need, and i posted all my attempts.

18. Sep 14, 2017

### Aows

i don't know how to solve it using FFCT @Orodruin

19. Sep 14, 2017

### Orodruin

Staff Emeritus
Yes, and you have been given help and guidance. That you are refusing to work with that guidance is up to you. Providing full answers is against the forum rules and you should not be expecting people to do so.

20. Sep 14, 2017

### Aows

Ok, then Mr. @Orodruin , can you tell me what is the first step ?