After seeing how we think about ordinary differential equations in chapter 1, we turn now to an
example of a partial differential equation, the heat equation.
To set things up, imagine you have some object like a piece of metal, and you know how the
heat is distributed across it at one moment; what the temperature of every individual point
is.
You might think of that temperature here as being graphed over the body.
The question is, how will that distribution change over time, as heat flows from the warmer
spots to the cooler ones.
The image on the left shows the temperature of an example plate with color, with the graph
of that temperature being shown on the right, both changing with time.
To take a concrete 1d example, say you have two rods at different temperatures, where
that temperature is uniform on each one.
You know that when you bring them into contact, the temperature will tend towards being equal
throughout the rod, but how exactly?
What will the temperature distribution be at each point in time?
As is typical with differential equations, the idea is that it’s easier to describe
how this setup changes from moment to moment than it is to jump to a description of the
full evolution.
We write this rule of change in the language of derivatives, though as you’ll see we’ll
need to expand our vocabulary a bit beyond ordinary derivatives.
Don’t worry, we’ll learn how to read these equations in a minute.
Variations of the heat equation show up in many other parts of math and physics, like
Brownian motion, the Black-Scholes equations from finance, and all sorts of diffusion,
so there are many dividends to be had from a deep understanding of this one setup.
In the last video, we looked at ways of building understanding while acknowledging the truth
that most differential equations to difficult to actually solve.
And indeed, PDEs tend to be even harder than ODEs, largely because they involve modeling
infinitely many values changing in concert.
But our main character now is an equation we actually can solve.
In fact, if you’ve ever heard of Fourier series, you may be interested to know that
this is the physical problem which baby face Fourier over here was solving when he stumbled
across the corner of math now so replete with his name.
We’ll dig into much more deeply into Fourier series in the next chapter, but I would like
to give at least a little hint of the beautiful connection which is to come.
This animation is showing how lots of little rotating vectors, each rotating at some constant
integer frequency, can trace out an arbitrary shape.
To be clear, what’s happening is that these vectors are being added together, tip to tail,
and you might imagine the last one as having a pencil at its tip, tracing some path as
it goes.
This tracing usually won’t be a perfect replica of the target shape, in this animation
a lower case letter f, but the more circles you include, the closer it gets.
This animation uses only 100 circles, and I think you’d agree the deviations from
the real path are negligible.
Tweaking the initial size and angle of each vector gives enough control to approximate
any curve you want.
At first, this might just seem like an idle curiosity; a neat art project but little more.
In fact, the math underlying this is the same as the math describing the physics of heat
flow, as you’ll see in due time.
But we’re getting ahead of ourselves.
Step one is to build up to the heat equation, and for that let’s be clear on what the
function we’re analyzing is, exactly.
The heat equation
To be clear about what this graph represents, we have a rod in one-dimension, and we’re
thinking of it as sitting on an x-axis, so each point of the rod is labeled with a unique
number, x.
The temperature is some function of that position number, T(x), shown here as a graph above
it.
But really, since this value changes over time, we should think of it this a function
as having one more input, t for time.
You could, if you wanted, think of the input space as a two-dimensional plane, representing
space and time, with the temperature being graphed as a surface above it, each slice
across time showing you what the distribution looks like at a given moment.
Or you could simply think of the graph of the temperature changing over time.
Both are equivalent.
This surface is not to be confused with what I was showing earlier, the temperature graph
of a two-dimensional body.
Be mindful of whether time is being represented with its own axis, or if it’s being represented
with an animation showing literal changes over time.
Last chapter, we looked at some systems where just a handful of numbers changed over time,
like the angle and angular velocity of a pendulum, describing that change in the language of
derivatives.
But when we have an entire function changing with time, the mathematical tools become slightly
more intricate.
Because we’re thinking of this temperature as a function with multiple dimensions to
its input space, in this case, one for space and one for time, there are multiple different
rates of change at play.
There’s the derivative with respect to x; how rapidly the temperature changes as you
move along the rod.
You might think of this as the slope of our surface when you slice it parallel to the
x-axis; given a tiny step in the x-direction, and the tiny change to temperature caused
by it, what’s the ratio.
Then there’s the rate of change with time, which you might think of as the slope of this
surface when we slice it in a direction parallel to the time axis.
Each one of these derivatives only tells part of the story for how the temperature function
changes, so we call them “partial derivatives”.
To emphasize this point, the notation changes a little, replacing the letter d with this
special curly d, sometimes called “del”.
Personally, I think it’s a little silly to change the notation for this since it’s
essentially the same operation.
I’d rather see notation which emphasizes the del T terms in these numerators refer
to different changes.
One refers to a small change to temperature after a small change in time, the other refers
to the change in temperature after a small step in space.
To reiterate a point I made in the calculus series, I do think it's healthy to initially
read derivatives like this as a literal ratio between a small change to a function's output,
and the small change to the input that caused it.
Just keep in mind that what this notation is meant to convey is the limit of that ratio
for smaller and smaller nudges to the input, rather than for some specific finitely small
nudge.
This goes for partial derivatives just as it does for ordinary derivatives.
The heat equation is written in terms of these partial derivatives.
It tells us that the way this function changes with respect to time
depends on how it changes with respect to space.
More specifically, it's proportional to the second partial derivative with respect to x.
At a high level, the intuition is that at points where the temperature distribution
curves, it tends to change in the direction of that curvature.
Since a rule like this is written with partial derivatives, we call it a partial differential
equation.
This has the funny result that to an outsider, the name sounds like a tamer version of ordinary
differential equations when to the contrary partial differential equations tend to tell
a much richer story than ODEs.
The general heat equation applies to bodies in any number of dimensions, which would mean
more inputs to our temperature function, but it’ll be easiest for us to stay focused
on the one-dimensional case of a rod.
As it is, graphing this in a way which gives time its own axis already pushes the visuals
into three-dimensions.
But where does an equation like this come from?
How could you have thought this up yourself?
Well, for that, let’s simplify things by describing a discrete version of this setup,
where you have only finitely many points x in a row.
This is sort of like working in a pixelated universe, where instead of having a continuum
of temperatures, we have a finite set of separate values.
The intuition here is simple: For a particular point, if its two neighbors on either side
are, on average, hotter than it is, it will heat up.
If they are cooler on average, it will cool down.
Focus on three neighboring points, x1, x2, and x3, with corresponding temperatures T1,
T2, and T3.
What we want to compare is the average of T1 and T3 with the value of T2.
When this difference is greater than 0, T2 will tend to heat up.
And the bigger the difference, the faster it heats up.
Likewise, if it’s negative, T2 will cool down, at a rate proportional to the difference.
More formally, the derivative of T2, with respect to time, is proportional to this difference
between the average value of its neighbors and its own value.
Alpha, here, is simply a proportionality constant.
To write this in a way that will ultimately explain the second derivative in the heat
equation, let me rearrange this right-hand side in terms of the difference between T3
and T2 and the difference between T2 and T1.
You can quickly check that these two are the same.
The top has half of T1, and in the bottom, there are two minuses in front of the T1,
so it’s positive, and that half has been factored out.
Likewise, both have half of T3.
Then on the bottom, we have a negative T2 effectively written twice, so when you take
half, it’s the same as the single -T2 up top.
As I said, the reason to rewrite it is that it takes a step closer to the language of
derivatives.
Let’s write these as delta-T1 and delta-T2.
It’s the same number, but we’re adding a new perspective.
Instead of comparing the average of the neighbors to T2, we’re thinking of the difference
of the differences.
Here, take a moment to gut-check that this makes sense.
If those two differences are the same, then the average of T1 and T3 is the same as T2,
so T2 will not tend to change.
If delta-T2 is bigger than delta-T1, meaning the difference of the differences is positive,
notice how the average of T1 and T3 is bigger than T2, so T2 tends to increase.
Likewise, if the difference of the differences is negative, meaning delta-T2 is smaller than
delta-T1, it corresponds to the average of these neighbors being less than T2.
This is known in the lingo as a “second difference”.
If it feels a little weird to think about, keep in mind that it’s essentially a compact
way of writing this idea of how much T2 differs from the average of its neighbors, just with
an extra factor of 1/2 is all.
That factor doesn’t really matter, because either way we’re writing our equation in
terms of some proportionality constant.
The upshot is that the rate of change for the temperature of a point is proportional
to the second difference around it.
As we go from this finite context to the infinite continuous case, the analog of a second difference
is the second derivative.
Instead of looking at the difference between temperature values at points some fixed distance
apart, you consider what happens as you shrink this size of that step towards 0.
And in calculus, instead of asking about absolute differences, which would approach 0, you think
in terms of the rate of change, in this case, what’s the rate of change in temperature
per unit distance.
Remember, there are two separate rates of change at play: How does the temperature as
time progresses, and how does the temperature change as you move along the rod.
The core intuition remains the same as what we just looked at for the discrete case: To
know how a point differs from its neighbors, look not just at how the function changes
from one point to the next, but at how that rate of change changes.
This is written as del^2 T / del-x^2, the second partial derivative of our function
with respect to x.
Notice how this slope increases at points where the graph curves upwards, meaning the
rate of change of the rate of change is positive.
Similarly, that slope decreases at points where the graph curves downward, where the
rate of change of the rate of change is negative.
Tuck that away as a meaningful intuition for problems well beyond the heat equation: Second
derivatives give a measure of how a value compares to the average of its neighbors.
Hopefully, that gives some satisfying added color to this equation.
It’s pretty intuitive when reading it as saying curved points tend to flatten out,
but I think there’s something even more satisfying seeing a partial differential equation
arise, almost mechanistically, from thinking of each point as tending towards the average
of its neighbors.
Take a moment to compare what this feels like to the case of ordinary differential equations.
For example, if we have multiple bodies in space, tugging on each other with gravity,
we have a handful of changing numbers: The coordinates for the position and velocity
of each body.
The rate of change for any one of these values depends on the values of the other numbers,
which we write down as a system of equations.
On the left, we have the derivatives of these values with respect to time, and the right
is some combination of all these values.
In our partial differential equation, we have infinitely many values from a continuum, all
changing.
And again, the way any one of these values changes depends on the other values.
But helpfully, each one only depends on its immediate neighbors, in some limiting sense
of the word neighbor.
So here, the relation on the right-hand side is not some sum or product of the other numbers,
it’s also a kind of derivative, just a derivative with respect to space instead of time.
In a sense, this one partial differential equation is like a system of infinitely many
equations, one for each point on the rod.
When your object is spread out in more than one dimension, the equation looks quite similar,
but you include the second derivative with respect to the other spatial directions as
well.
Adding all the second spatial second derivatives like this is a common enough operation that
it has its own special name, the “Laplacian”, often written as an upside triangle squared.
It’s essentially a multivariable version of the second derivative, and the intuition
for this equation is no different from the 1d case: This Laplacian still can be thought
of as measuring how different a point is from the average of its neighbors, but now these
neighbors aren’t just to the left and right, they’re all around.
I did a couple of simple videos during my time at Khan Academy on this operator, if
you want to check them out.
For our purposes, let’s stay focused on one dimension.
If you feel like you understand all this, pat yourself on the back.
Being able to read a PDE is no joke, and it’s a powerful addition to your vocabulary for
describing the world around you.
But after all this time spent interpreting the equations, I say it’s high time we start
solving them, don’t you?
And trust me, there are few pieces of math quite as satisfying as what poodle-haired
Fourier over here developed to solve this problem.
All this and more in the next chapter.
I was originally inspired to cover this particular topic when I got an early view of Steve Strogatz’s
new book “Infinite Powers”.
This isn’t a sponsored message or anything like that, but all cards on the table, I do
have two selfish ulterior motives for mentioning it.
The first is that Steve has been a really strong, perhaps even pivotal, advocate for
the channel since its beginnings, and I’ve had the itch to repay the kindness for quite
a while.
The second is to make more people love math.
That might not sound selfish, but think about it: When more people love math, the potential
audience base for these videos gets bigger.
And frankly, there are few better ways to get people loving the subject than to expose
them to Strogatz’s writing.
If you have friends who you know would enjoy the ideas of calculus, but maybe have been
intimidated by math in the past, this book really does an outstanding job communicating
the heart of the subject both substantively and accessibly.
Its core theme is the idea of constructing solutions to complex real-world problems from
simple idealized building blocks, which as you’ll see is exactly what Fourier did here.
And for those who already know and love the subject, you will still find no shortage of
fresh insights and enlightening stories.
Again, I know that sounds like an ad, but it’s not.
I actually think you’ll enjoy the book.