Hello.

Question 1

Find the temperature distribution T(x,t) in a long thin bar $\displaystyle -a \leq x \leq a$ with given initial temperature $\displaystyle 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 $\displaystyle T=0$.

The radiation at the ends is taken to obey Newton's Law.

In particular, find the Fourier coefficients in terms of $\displaystyle f(x)$

Here's what I did so far:

I set the boundary condition to be $\displaystyle u(-a) = 0, u(a) = 0$, initial condition to be $\displaystyle u(x,0) = f(x)$.

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

$\displaystyle u_{tt} + 2au_t + bu - c^2u_{xx} = 0$

where $\displaystyle a, b, c$ are positive constants.

Solve by separation of variables

$\displaystyle u_{tt} + 2au_t + bu - cu_{xx} = 0, 0 < x < \pi, t > 0$

$\displaystyle u(0,t) = u(\pi, t) = 0, t > 0$

$\displaystyle u(x,0) = 0, u_t(x,0) = g(x)$

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

And again here's what I did:

$\displaystyle u = X(x)T(t)$

$\displaystyle XT'' +2aXT' + bXT - c^2X''T =0$

$\displaystyle X(T'' + 2aT' + bT) = c^2X''T$

$\displaystyle \frac{T'' + 2aT' + bT}{c^2T} = \frac {X''}{X}$

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

Thank you very much.