8.1 Introduction

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I shall assume that you have encountered Ordinary Differential Equations or

ODE’s for short, and that you have grasped that they are central to much

of the science and technology that have changed the world in the last few

centuries. Now you get to find out why they are called ‘Ordinary’.

A crucial feature of setting up an ODE or a system of ODE’s is that we have

two elements. One is the local law of dynamics which says very generally how

things tend to change locally. The other is the boundary condition, often an

initial value telling where something starts. Putting these together, we get a

‘solution’ that is, say, the particular time development of a system.

There is an obvious generalisation of ODEs to the situation where instead

of something varying in just one dimension, time in many cases, it can vary

in two (or more) dimensions. A solution to an ODE is a curve (usually

the path in state space of some system as time changes). A solution to

the more general problem and to a Partial Differential Equation, PDE for

short, would be some surface (or higher dimensional manifold) sitting in a

space. We expect the crucial two features to remain the same: there will be

some local law for the system and some boundary conditions which select a

particular solution from an infinite family of them.

Example 8.1.1. I take a loop of wire and twist it about a bit. Then I dip

it in soap solution and get a nice soap film in the wire. If I hold the wire up

in R3, there is a function defined over the ‘shadow’ of the loop and the film,

which tells us the height of the soap film everywhere. More generally, I have

157

158 CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS

Figure 8.1: Blowing Bubbles.

a function from S1 into R3 which embeds the circle in three-space, and this

extends to a function from the unit disk, D2 into R3.

The illustration of figure 8.1 shows you the possibilities.

The soap film extension is only one among an infinite number of possible

extensions (blow on the film to distort it to get some others), the question is,

what made the film choose the particular shape it did? The answer is that

surface tension was busy trying to minimise the area, given the boundary.

Now this is a purely local thing, like a vector field, while the surface that

you actually get is a global solution. The shape of the boundary wire is the

boundary condition. So there ought to be a way of setting up something that

is a generalisation of an ODE and finding a way to solve it which would give

a solution to the soap film problem.

There is indeed a whole body of Mathematics dedicated to precisely this sort

of problem and its higher dimensional analogues, and it is called the study of

Partial Differential Equations. Just as ordinary differential equations have

differentiation of the time or some other single variable because the solution

is a curve, so the PDEs have partial derivatives occurring in them because

the solution will be a surface or some higher dimensional manifold. It is more

complicated than ODE theory for several reasons, one of them being that the

boundary of a curve is just a pair of points (unless the curve is closed, when it

doesn’t have a boundary), whereas the boundary of a two dimensional thing

like a disk is a circle, which is a lot more complicated than a couple of points.

Actually, most PDEs are so hard we don’t have the foggiest ideas about how

8.1. INTRODUCTION 159

to solve them1, we can only do a few easy ones. But those we can solve are

very, very, useful. In the remainder of the course I can only start on the

subject, but I shall try to see that you get a feel for the basics.

Example 8.1.2. I take a solid ball of iron and sit it on a table. Everything

is at room temperature. Now I heat up the table just under the ball by

applying a blow-torch, the temperature of which is rather a lot higher than

room temperature, say 10000. How does the temperature of the interior

point (x, y, z) of the solid ball change in time? It obviously starts off at

room temperature at time zero, and then goes up fairly fast, and the closer

(x, y, z) is to the blow torch, the faster it goes up. It would be nice to

have some details: leaving it to the fluffiness of natural language is not good

enough for scientists. The answer would be a function of four variables,

x, y, z and t. If we could obtain such a function and confirm its correctness

by experimenting, we should undoubtedly feel we understood a fair bit about

heat flow, something which could come in useful.

Remark 8.1.1. If I have a function f : R2 −! R, and if I differentiate it, I

get a (row) matrix of partial derivatives,

_

@f

@x

,

@f

@y

_

It makes sense therefore to guess that if there is a (Partial) differential equation

the solution to which is a disk or ball mapped into R, the the equation

itself will have partial derivatives in it. Hence the name.

Example 8.1.3. It can be shown that if f : D2 _ R2 −! R is a function

which describes the height of a soap film above the z = 0 plane, then provided

there are no other forces but surface tension operating, and providing the

function f on the boundary is not too different from a constant function,

then f approximately satisfies the condition

@2f

@x2 +

@2f

@y2 = 0

or

fxx + fyy = 0

if this notation is more to your taste.

1Well, closed form or analytic solutions in terms of standard functions are very rare,

and even solutions in terms of explicit infinite series are often impracticable. But numerical

methods can give us a solution to high accuracy in many cases. Determining whether the

numerical solution is a stable, safe one is still under investigation.

160 CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS

If you are told that f : S1 −! R is a particular function, and that

@2f

@x2 +

@2f

@y2 = 0

then the instruction ‘solve for f’ means to find the unique (you hope) function

f : D2 −! R which has this property and is as specified on S1 = @D2.

Remark 8.1.2. PDE problems where we know relationships such as this

locally, and we are given all values of a function on a boundary and want

to find the function on the interior, are called Dirichlet Problems after the

man who made a speciality of tackling them in the nineteenth century. This,

incidentally, was the first bloke to work out what a function is. The function

evil f which is 1 on the irrationals and -1 on the rationals was his idea. He

was German, not French as the name suggests. He was born in 1805, so this

is all recent stuff, only a century and a half or so old2.

Another kind of problem, not a Dirichlet problem but related is:

Example 8.1.4. If I heat up one half of a copper rod to 100o and keep the

other half at 0o while doing so (try not to think about the midway point) and

then take away the freezer and the flame, the function of length giving the

temperature will start off as a step function and gradually even out until the

bar is a nice 50o everywhere, assuming no heat is lost to the outside world.

Given information about how heat is conducted through the material, we

ought to be able to compute the function at any time after t0. We think

of time as the positive reals, so we know the value of the function _(x, t)

completely at t = 0, the step function, where _ is the temperature, and it is

known that the Heat Equation must be satisfied:

@_

@t

= c2 @2_

@x2

So again we have a partial differential equation.

Partial Differential Equations then occur quite naturally as ways of describing

Physical systems. We have two jobs to do:

• From a physical situation, set up the equation which describes the

system

2You may reasonably suspect that this is a joke. On the other hand, most of what

you did in first year was known to Newton in 1695 when he had more or less given up on

Science and Mathematics as less important than Theology. The first artificial satellite had

been invented by Newton many years before. It took the Engineers about three hundred

years to catch up. Seen from that point of view, you are doing quite well.

8.2. THE DIFFUSION EQUATION 161

• Solve the equation

We next consider some simple cases of the first part, setting up the equation.

I shall defer considering how to actually solve them for some time. The cases

we look at will be very simple and may give you the completely erroneous

impression that mathematicians are interested only in simple things like heat

conduction along a rod and vibrating strings. The reason we are looking at

simple cases is essentially the same reason as you do not give a three month

old baby a nice steak dinner to eat. It has neither the teeth to bite into it nor

the digestion to absorb it. So you give it squishy stuff instead that doesn’t

need big teeth or a strong jaw. Once you have shown you can chomp through

the easy cases, you will be ready to chew on the more interesting problems.

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