The discrete heat problem - Part 0: Introduction

Newton's law of cooling states that "the rate of heat loss is proportional to the difference in temperatures between the body and its surroundings". Two years ago I began playing around with Newton's idea (although at the time I did not know it was attributed to him). The problems that have arisen since are the most rewarding ones that I have come across in my lifetime, and now, I'd like to share them with you.

I'll start with the one which I call the discrete heat problem: Consider the points \((i, 0) (i \in \mathbb{Z}; 0 \leq i < n; 2 \leq n)\). Each has an amount of heat, which we'll denote \(E_i(t)\) as a function of time. Suppose we are given \(E_i(0) (0 \leq i < n)\). Can we find explicit expressions for \(E_i(t)\)?

The problem can be solved one part at a time, each part being interesting on its own, so I have decided to split the problem into several posts. Here's an outline of my solution:

1. Create a differential equation
2. Solve the characteristic equation
3. Create an associated linear equation (almost done!)
4. Solve the linear equation (not done yet!)