How do I discretize a parabolic partial differential equation?

I have the following homework question:

To keep my long sob story as short as possible, my awesome applied numerical methods teacher had a personal emergency and is replaced for the rest of the semester. Let's just say I find my new professor's teaching style to be a mismatch to my learning style.

Anyways, I have pages and pages of notes on parabolic PDE's, but they simply make no sense to me as there was no organization to them.

The professor just basically spews anything that pops into his head and it's hard to tell what is important and what is not.

PLEASE DO NOT GIVE ME THE ANSWER, THAT WON'T HELP ME. What will help me is either an explanation or a link to a video/other source that makes this homework problem make some sense. I'm just looking for a step in the right direction. Once I understand this first question I'll easily be able to implement it in MATLAB for the followup questions.

Can someone give me some direction as to how to begin this problem?

The usual method is to replace derivatives with difference quotients evaluated at the appropriate grid points. For example, you might write $\frac{df}{dx}$ as
$\displaystyle \frac{f(x_i) - f(x_{i-1})}{x_i - x_{i-1}}$, etc.
• Since it's a partial derivative, you may want to keep the $x$-coordinate fixed. – Tom Mar 19 '15 at 4:03