# Heat Equation - B.Cs

1. Jul 17, 2009

### 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

#### Attached Files:

• ###### Heat Equation - cylindric coordinates - only radial dependance.doc
File size:
15 KB
Views:
43
2. Jul 17, 2009

### 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

#### Attached Files:

• ###### lpde105.pdf
File size:
53.7 KB
Views:
69
3. Jul 22, 2009

### 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

4. Jul 22, 2009

### HallsofIvy

Staff Emeritus
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.

5. Jul 23, 2009

### muzialis

HallsofIvy,

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

6. Jul 23, 2009

### HallsofIvy

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

7. Jul 23, 2009

### 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

8. Jul 30, 2009

### 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