How to Solve a Heat Equation Using FFCT?

[Orodruin]

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.

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.

 

[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.
so that's why am asking for the answer...

 

[Aows]

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

 

[Orodruin]

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.
so that's why am asking for the answer...

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.

 

[Aows]

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

 

[Orodruin]

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

I already did.

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 $x=L$?

 

[Aows]

I already did.

I will try my best to follow this step even though this is the first time using it, hopefully we reach to a solution.
minutes and i will show you my progress @Orodruin

 

[Aows]

so, this first problem is to find the h(x) right? if so, how to find it ? Mr. @Orodruin

 

[Aows]

@TSny

 

[Aows]

@Orodruin , any ideas on how to find the $h(x)$ ?

 

[Orodruin]

As I said already, it is the stationary solution to the problem. What is particular about the stationary solution?

 

[Aows]

what is the meaning of stationary solutions ? and how to find it ?? @Orodruin

 

[Orodruin]

These are really basic questions that should be covered in your textbook. A stationary solution is a time-independent solution.

 

[Aows]

Mr. @Orodruin , please excuse my basic questions for the moment, just put yourself in my shoes, exam is two days away.
the definition of the stationary solutions is clear now, but how to find it so that i can then use it here $u(x,t) = v(x,t) + h(x)$ ?

 

[Orodruin]

just put yourself in my shoes, exam is two days away.

This has nothing to do with how we offer to help people here. You created this thread over two weeks ago and until now you have shown very little interest in putting in the effort necessary to actually solve the problem. We are volunteers and mostly help people who want to be helped. Stating that you need help immediately because your exam is two days away is likely to have the exact opposite effect as compared to what you are going for. In order to learn this properly, you need to sit down with the material and think about each step. I have already given you several hints that should be sufficient to at least find the stationary solution.

You are saying that you know understand what a stationary solution is, but I have my doubts because it seems that you put exactly zero effort into thinking about what this means and until you do so you will not be able to really learn the subject. Just seeing a lot of examples will not get you far, you need to sit down to think about and understand what the meaning behind what we are telling you is. Since it does not seem that we are going in that direction, I am done with this thread.

 

[Aows]

but Mr. @Orodruin , when i asked about the mathematical meaning of the stationary solutions you only replied with a definition. and i need to know how to apply it mathematically, any hints about how to apply it ?

 

[Jorgen1224]

Sorry for asking it here, but i wanted to know what are these FFCT and FFST? Is it like some special method or something related to this exercise? Can't find it in the internet.

 

[TSny]

FFCT is "finite Fourier cosine transformation". Likewise, FFST is "finite Fourier sine transformation". See for example http://www.math.usm.edu/lambers/mat417/lecture18.pdf

"finite" just refers to the fact that the range of integration in the transformation is finite.