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Thursday, April 3, 2014

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)(iZ;0i<n;2n). Each has an amount of heat, which we'll denote Ei(t) as a function of time. Suppose we are given Ei(0)(0i<n). Can we find explicit expressions for Ei(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!)

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