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Differential equations

A childhood question about heat

by John on March 10, 2010

When I was a little kid, I asked some adults the following question.

If hot things cool, and cool things warm up, could something hot cool down and warm back up?

The people I asked didn’t understand my question and just laughed. I have no idea how old I was, but I wasn’t old enough to articulate what I was thinking.

Here’s what I had in mind. I knew that hot things like a cup of coffee grew cold. And I knew that cold things, say a glass of milk, get warm. Well, could the coffee get so cold that it becomes a cold thing and start to warm back up?

Could the coffee become as cold as the glass of milk? Common sense suggests that can’t happen. When we say coffee grows cold, we mean that it becomes relatively colder, closer to room temperature. And when we say the milk is getting warm, we also mean it is getting closer to room temperature. We’ve never left a hot cup of coffee on a table and come back later to find that it has cooled off so much that it is colder than room temperature. But could there be small fluctuations?

As the coffee and milk head toward room temperature, could they overshoot the target, just by a little bit? Say room temperature is 70 °F, the coffee starts out at 150 °F, and the milk starts out at 40 °F. We don’t expect the coffee to cool down to 40 °F or the milk to warm up to 150 °F. But could the coffee cool down to 69.5 °F and then go back up to 70 °F? Could the milk warm up to 70.5 °F and then cool back down to 70 °F?

I didn’t get a satisfactory answer to my childhood question until I was in college. Then I found out about Newton’s law of cooling. It says that the rate at which a warm body cools is proportional to the difference between its current temperature and the ambient temperature. This law can be written as a differential equation whose solution shows that the temperature of a warm body decreases exponentially to the ambient temperature. The temperature curve always slopes downward. It doesn’t wiggle even a little on its journey to room temperature. Cold bodies warm up the opposite way, exponentially approaching room temperature but never exceeding it.

In case it this seems obvious, think about thermostats. They don’t work this way. Say the temperature in a room is 85 °F and you’d like it to be 72 °F, so you turn on the air conditioning. Will the temperature steadily lower to 72 °F? Not exactly. If you were to plot the temperature in the room over time and look at the graph from far enough away, it would look like it is steadily going down to the desired temperature. But if you look at the graph more closely, you’ll see wiggles. The AC may cool the room to a little below 72 °F, maybe to 70 °F. The AC would cut off and the temperature would rise to 72 °F. Unlike the cup of hot coffee, the AC will often overshoot its target, though not by too much. The temperature may feel constant, but it is not. It oscillates around the desired temperature.

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How to differentiate a non-differentiable function

by John on October 25, 2009

How can we extend the idea of derivative so that more functions are differentiable? Why would we want to do so? How can we make sense of a delta “function” that isn’t really a function? We’ll answer these questions in this post.

[click to continue...]

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Approximating a solution that doesn’t exist

by John on August 11, 2009

The following example made an impression on me when I first saw it years ago. I still think it’s an important example, though I’d draw a different conclusion from it today.

Problem: Let y(t) be the solution to the differential equation y’ = t2 + y2 with y(0) = 1. Calculate y(1).

If we use Euler’s numerical method with a step size h = 0.1, we get y(1) = 7.19. If we reduce the step size to 0.05 we get y(1) = 12.32. If we reduce the step size further to 0.01, we get y(1) = 90.69. That’s strange. Let’s switch over to a more accurate method, Runge-Kutta. With a step size 0.1 the Runge-Kutta method gives 735.00, and if we use a step size of 0.01 we get a result larger than 1015. What’s going on?

The problem presupposes that a solution exists at t = 1 when in fact no solution exists. General theory (Picard’s theorem) tells that a unique solution exists for some interval containing 0, but it does not tell us how far that interval extends. With a little work we can show that a solution exists for t at least as large as π/4. However, the solution becomes unbounded somewhere between π/4 and 1.

When I first saw this example, my conclusion was that it showed how important theory is. If you just go about numerically computing solutions without knowing that a solution exists, you can think you have succeeded when you’re actually computing something that doesn’t exist. Prove existence and uniqueness before computing. Theory comes first.

Now I think the example shows the importance of the interplay between theory and numerical computation. It would be nice to know how big the solution interval is before computing anything, but that’s not always possible. Also, it’s not obvious from looking at the equation that there should be a problem at t = 1. The difficulties we had with numerical computation suggested there might be a theoretical problem.

I first saw this problem in an earlier edition of Boyce and DiPrima. The book goes on to approximate the interval over which the solution does exist using a combination of analytical and numerical methods. It looks like the solution becomes unbounded somewhere near t = 0.97.

I wouldn’t say that theory or computation necessarily come first. I’d say you iterate between them, starting with the approach that is more tractable. Theoretical results are  more satisfying when they’re available, but theory often doesn’t tell us as much as we’d like to know. Also, people make mistakes in theoretical computation just as they do in numerical computation. It’s best when theory and numerical work validate each other.

The problem does show the importance of being concerned with existence and uniqueness, but theoretical methods are not the only methods for exploring existence. Good numerical practice, i.e. trying more than one step size or more than one numerical method, is also valuable. In any case, the problem shows that without some diligence — either theoretical or numerical — you could naïvely compute an approximate “solution” where no solution exists.

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