Heat Equation B.Cs: Solve & Determine Coefficients

[muzialis]

Hello there,

hope you are having a good one.

My problem is to solve the heat equtaion in cylindrical coordinates.
This has been done by others for me, so a closed form solution is available, please see attached (please note the problem is 1 - D due to initial conditions depending only on r).



My b.c. are as follows:
For all time, fixed temperature at the inner and outer radius of the hollow cylinder
At time= 0, all the cyclinder is uniformly at the same tempeature as the outer radius.

Using these conditions I should determine the coefficients Am and c in the solution, but I am unsure on how to do in relation to my b.c.s.
At time 0 my b.c. are discontinuous, which I am unable to replicate by any choice of the coefficients. Still I am sure they are legal, as I dealt long time with other cases where the initla temperature distribution was the dirac function.

Any hint would be the most appreciated, thank you very much

Muzialis

 

[muzialis]

Its me again.

I just forgot to mention I also found this other source, which I attach, about solving the PDE using Green's functions, which would cope with the discontinuous b.c. very nicely.

But I can not understand why there are two different Green's functions for two different boundary value problems. I am all but on expert on this, still i was under the impression every linear operator would have its own Green function, full stop.

Any explanation would be so welcome as well.

All the Best

Muzialis

 

[muzialis]

All,

I do hate being so molest, but still I was wondering if anybody had any help to give.

It would be relaly appreciated.

Thank you

Best Regards

Muzialis

 

[HallsofIvy]

Is this problem "interior" or "exterior"? That is, is your equation to be satisfied inside or outside the cylinder?

If inside, then an additional condition is that it be continuous at r= 0.

If outside, then an additional condition is that it must not "blow up" (go to infinity) as r goes to infinity.

 

[muzialis]

HallsofIvy,

thank you for your post.

The equation is to be satisfied in the region a < r < b. The problem is moreover centrally simmetric.
Still I am unsure on how to proceed.

Kindest Regards

Muzialis

 

[HallsofIvy]

Then you should have boundary conditions on the two circle r= a and r= b.

 

[muzialis]

HalfsofIvy,

many thanks for youe elucidation.

I think I understand what you say. Actually in my first post i describe the boundary conditions as

"My b.c. are as follows:
For all time, fixed temperature at the inner and outer radius of the hollow cylinder
At time= 0, all the cyclinder is uniformly at the same tempeature as the outer radius <..>".

My probelm though starts here.

All the best and many thanks for your time

Muzialis

 

[muzialis]

Hi All,

well, it is me again.
Let me apologize for my stubbornness, still as they say, asking is never impolite.

could anybody forgive my insistence and provide me with a clue on my original post?

Thank you again and sincerely sorry for having to be this molest

All the Best

Muzialis