Partial Differential Equations Problems

Hello.

**Question 1**

Find the temperature distribution T(x,t) in a long thin bar with given initial temperature

The side walls of the bar are insulated, while heat radiates from the ends into the surrounding medium whose temperature is .

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

In particular, find the Fourier coefficients in terms of

**Here's what I did so far:**

I set the boundary condition to be , initial condition to be .

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

where are positive constants.

Solve by separation of variables

For simplicity assume that .

**And again here's what I did:**

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

Thank you very much.