Partial Differential Equations Problems

**Creebe** · September 21st 2010, 09:54 AM

Hello.

Question 1
Find the temperature distribution T(x,t) in a long thin bar $-a \leq x \leq a$ with given initial temperature $u(x,0) = f(x)$
The side walls of the bar are insulated, while heat radiates from the ends into the surrounding medium whose temperature is $T=0$ .
The radiation at the ends is taken to obey Newton's Law.
In particular, find the Fourier coefficients in terms of $f(x)$

Here's what I did so far:
I set the boundary condition to be $u(-a) = 0, u(a) = 0$ , initial condition to be $u(x,0) = f(x)$ .

I am not sure what to do with the "radiation" part of the problem, and thus i don't really know what my partial differential equation should look like...

Question 2
A wave system that includes damping and dispersion is represented by
$u_{tt} + 2au_t + bu - c^2u_{xx} = 0$
where $a, b, c$ are positive constants.
Solve by separation of variables
$u_{tt} + 2au_t + bu - cu_{xx} = 0, 0 < x < \pi, t > 0$
$u(0,t) = u(\pi, t) = 0, t > 0$
$u(x,0) = 0, u_t(x,0) = g(x)$

For simplicity assume that $a^2 - b - c^2 < 0$ .

And again here's what I did:

$u = X(x)T(t)$
$XT'' +2aXT' + bXT - c^2X''T =0$
$X(T'' + 2aT' + bT) = c^2X''T$
$\frac{T'' + 2aT' + bT}{c^2T} = \frac {X''}{X}$

Please help...and in the mean time, I'll try to work something out with my classmates...

Thank you very much.

**Ackbeet** · September 21st 2010, 10:28 AM

Question 2:

You're doing great so far. What's the next step? You've got all $T$ on the LHS, and all $X$ on the RHS. What can you say?

**Creebe** · September 21st 2010, 05:20 PM

I can say that I have separated the variables. Should I make the equation equal to $\lambda$ ?

And I'm not too sure what to do with the $T$ part.

I know what to do for X now.

Nvm...i get it now. This problem isn't that hard after all...probably because I had a limited lunch break.

Still working on Question 1 though.

**Ackbeet** · September 22nd 2010, 05:13 AM

Glad you got Question 2. As for Question 1, I'm not so sure your boundary conditions are correct. You're supposed to assume Newtonian cooling there, not necessarily equilibrium with ambient temperature. So I'd say your initial conditions would be $u(-a,0)=f(-a),$ and $u(a,0)=f(a).$ The values at the endpoints then obey Newton's Law of Cooling compared with the ambient environment. Perhaps Danny or CB could correct me if I'm wrong, but that's how I read it. Not sure how you're going to get the heat transfer coefficient. I suppose you could always just assume it's a constant, and have it show up in your final answer.

Those are my thoughts.

**Danny** · September 22nd 2010, 03:04 PM

Following from Ackbeet, from what I see, your radiating BC at the endpoints is

$u_x = - k\left( u - u_0\right)$ where $u_0 = 0$ .

**Ackbeet** · September 22nd 2010, 04:22 PM

Or would that be

$u_{t}=-k(u-u_{0})=-ku$ ?

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