Okay. Good morning, everybody.
We're now starting partial differential equations, which is certainly one of the richest topics that humanity has ever created.
I think you could say we've been talking about Odie now.
PD So. PD Partial differential equation, which means at least two independent variable.
So to begin with, a bit of the history, of course, Newton and Live, MIT and so on invented calculus.
And that was in the second half of the 17th century. And I guess from there, in the ensuing hundred and 50 years or so,
people did more and more with differential equations and then partial differential equations, especially linear ones.
So in the 19th century, you would have had a huge understanding of linear partial differential equations.
And then in the 20th century, non linearity became important.
So now we have this very rich world of linear and nonlinear problems.
Roughly speaking, in the linear case, you can often say more theoretically.
In the nonlinear case you can do amazing things computationally,
and many important things on earth are linear and many other important things are nonlinear.
They're both very important. I brought along a couple of books.
There are of course, hundreds of books on Pdes.
I brought along a couple of my favourites, which from your point of view would be sort of at the mathematics end of things.
These are certainly not science or engineering books. One is by far with a lurid cover introduction to partial differential equations.
And another classic, which keeps getting reissued every decade or so, is Fritz John Partial Differential Equations.
So let's begin with the standard notation that we'll use and that people use all the time, namely subscripts.
When you write you sub t in this world that generally means the partial derivative of you with respect to t.
And if you write you sub x, of course it's the partial with respect to x.
And if you write multiple subscripts, the obvious thing happened like u x x would be the partial squared of you with respect to x squared.
So with notation like that, you can write down PDS compactly in particular.
There's also some vector notation which we know and love the gradient and the Laplace.
So when I write that symbol, I mean the gradient. So the gradient of a scalar function u is a vector in three dimensions.
It would be the vector of the three partial derivatives. You X, you y, you z.
And numerical people generally think of vectors as columns.
So I would tend to think of that as a column vector. And similarly so that's the gradient.
The other big one is the Laplace, in which we write with an upward pointing triangle, and that's a scalar,
namely the sum of the second derivatives in three D that would be u, x, x, plus u, y, y plus u z.
So that's c la plus in. These go back about 200 years to the glory days of French mathematics after Napoleon.
In the handouts there for handouts. You'll notice a couple of pages from the PD coffee table book.
That was a project we did jointly about 15 years ago here at Oxford.
It never got finished, so we finished about 40 pages intending to do about 100.
So it has never been published, but it's freely available online.
If you go to my web page and look under books, you'll find it.
In each case, there's this two page spread describing a PD, and we tried to make it really beautiful and informative.
So although some years have passed, these are still the easiest way to get to know some of the world's interesting PD.
And, well, there are two of them today.
And I will continue to hand out a couple more in each lecture from now on so you can get to know some of our favourite PBDEs.
Let me write down a couple of standards. So.
For example, the LE Plus equation.
Is that the plus in of a function equals zero?
And that makes sense in 3D, but also in 2D. And one day it would be a trivial, odd EU x x equals zero.
People classify PDS into different categories, which I won't really go into very much,
but this one is the classic, the canonical example of an elliptic PD.
The Poisson equation is the same thing, except with a right hand side that may be non-zero.
So that would be LA plus the interview equals F where F would be in general a function of the space variables.
So that's also elliptic. Intuitively, an elliptic equation is one where everything is coupled together instantly.
There's no no notion of time propagation.
The heat equation, also known as the diffusion equation, now does have a time variable.
So that would be u t equals le plus the end of u.
So that now is a time dependent equation.
It's classified, this one as parabolic, which loosely speaking, means a time dependent PDT where information travels infinitely fast.
And then the canonical example of the third category would be the wave equation.
Which is the second order version of this u t t equals le plus ian.
So this is all second derivative duty equals u x x plus u y y plus u.
That's the canonical example of the hyperbolic PD.
And loosely speaking, a hyperbolic equation is one where information travels at a finite speed.
So for this equation, the way I've written it, information travels at speed.
One, just to give a couple examples of a non-linear equation.
There's the equation due to a guy called burgers.
So the burgers equation and that is u t equals u squared sub x plus u x x.
So there you see non linearity. You have a derivative of u squared.
That's the nonlinear term coupled with a linear and diffusive term.
I'm not putting any constants in these equations. Of course, in an application you'd have physical constants all over the place.
Then one other example of a famous nonlinear equation would be the CDV equation that stands for court to Vagaries.
And that's you t equals you squared x plus u x x x.
Little difference is like adding or subtracting an x of course have a huge effect.
When you have a skilled y, you instantly see a term like that is diffusion.
Which means things get smoother and sort of move off to infinity.
A term like this means dispersion. Which means that something like this turns into a wave train.
Very different behaviour. So diffusion is smooth. Dispersion of waves.
Let's look at the handout called examples of. Just to give you a sense of how vast the intellectual terrain here is, how much of science,
especially science but also engineering is related to these things and not just related, but founded on PBDEs.
It's absolutely at the heart of the intellectual world.
So I'll just read them through. So we have a sense of them.
The wave equation I've mentioned, and that would, for example, describe sound waves in air or water.
So the field of acoustics is based on the wave equation.
The heat equation describes diffusion of all sorts of things.
In the case of heat, it's momentum, it's the fusing. But of course other things can diffuse, like salt or whatever you like.
The LE Plus equation is the steady state version of these things.
So often you find that if there's no T in the equation, that's because tea has gone to infinity.
If you have a steady state solution of the heat equation, it's no longer changing in time.
So you have the LE Plus equation. So that's steady state and electromagnetics.
Electrostatic electromagnetics is very much built on that as the the first pillar,
the Laplace equation, the bi harmonic equation is the fourth order analogue of the LE Plus equation.
So the LE Plus equation is second order by harmonic.
Harmonic as LE Plus by harmonic is sort of harmonic squared.
That's a fourth order linear PD and that's the basis of solid mechanics.
Elastic waves are by harmonic, whereas acoustic waves are harmonic.
So acoustic is second order. Elastic is fourth order.
Turns out it's the same equation for this gives fluids in the limit of zero reynolds number where a viscosity completely dominates momentum.
The Poisson equation is a steady state. Potentials, as I say here, when you have sources.
So if you have an electrostatic field, for example, with some charge distributed around, then you'd have the LE Plus equation.
Plus on the question. Sorry, I guess I already mentioned the elastic wave equation.
The bi harmonic is a scalar version. Elastic waves actually involve two coupled fields.
So you have sound waves and solids actually involve pressure waves going this way and shear waves going that way.
So it's a coupled system of equations. The Helmholtz equation is what you get if you do acoustics, but you fix the frequency.
So if you look at the response of a system to an input at a given frequency, that becomes the Helmholtz equation.
Another Elliptic PD. Maxwell's equations.
Well, they're pretty famous. They were, I think, the first great example of a scientific advance that was really necessarily cast in the light of PBS.
This came long after Pdes existed. 50 years after Laplace, ampersand and all that stuff.
But when Maxwell did his work, you might know a bit of this history.
Initially, for a decade or so, he was working with fairly physical models of how electric,
how Faraday's discoveries might work, how electric fields and magnetic fields might interconnect.
But as time went by, he realised that the right way to do this was as a partial differential equation.
And his great paper, which created electromagnetic wave theory, wrote down the pdes that we call Maxwell's equations.
So that's linear. It's epical in impact, but linear.
And then Einstein's equations are very, very nonlinear. This is a coupled system of about ten nonlinear equations at the heart of general relativity.
If you look at a typical applied mathematician like me, I more or less know Maxwell's equations,
and I certainly don't really know Einstein's equations. Is there anyone in the room who works with Einstein's equations?
Okay. One of the great discoveries of all time. Schrodinger's equation is the basis of quantum mechanics and chemistry, if you like.
I personally think that Schrodinger's equation was the most important scientific advance of the 20th century.
With the date DNA, Watson and Crick would come in as number two because once we had this equation in principle, the whole periodic table is explained.
Chemistry suddenly has a mathematical basis.
It's amazing the impact of Schrodinger's equation.
The court of negative freeze equation I mentioned here the characteristic solutions of those are solid times.
These are nonlinear waves which travel at a speed that depends on their amplitude and interact in interesting ways.
The the equation dates to the early 20th century,
but it was in the 1970s that people discovered solid lines and that really for a decade became the centrepiece of applied mathematics.
The Klein Gordon equation is something that comes up in quantum mechanics attempts to relate that to relativity.
Berger's equation I mentioned is at the heart of the theory of shockwaves.
If you don't have this term there at all, then you get pure shock waves with with absolute discontinuity.
If you have a little viscosity in there, you get smoothed off shock waves.
So typically there'd be a small epsilon multiplying this term in a Berger's equation as another.
Your Stokes equations are the basis of all the fluid mechanics.
So you take any Newtonian fluid, the navier-stokes equations which date to 1846 or so describe how it evolves.
And really, for the last hundred and 70 years we've been trying to solve them in various configurations.
The Euler equations are the same thing, but without viscosity, so they're a special case of the Navier-stokes equations.
When viscosity doesn't matter, the fisher CP equation, which we'll get to at the end of today, is a simple travelling wave equation.
So it describes the sort of thing like pulses in neurones.
It actually doesn't do precisely that, but it's that kind of flavour.
The next one, the Hodgkin Huxley equation, is precisely more complicated system of equations designed to model neurones.
The black holes equation is something that came up a few decades ago for valuation of options in finance.
I remember when I first heard about it, it seemed like a silly gimmick, but now it's infinitely important.
So virtually all of what I've mentioned is science. But then there are also equations that come up in engineering.
In other words, equations that people design in order to achieve something.
And certainly the paranormal like equation is in that category.
It's a nonlinear diffusion equation designed so that instead of smoothing away edges actually sharpen up in an interesting fashion.
And that is one of the early ones in a whole industry of using PDA used to do image processing.
The Koshy Riemann equations are an elliptic pair of equations which are at the heart of complex analysis of the Allen Khan equation,
which will play with a bit related to formation of structure in alloys and other materials.
The key promoter efficiency equation is a beautiful example of a chaotic problem.
So it's a PDA whose solutions are chaotic.
They look chaotic on the screen, and they can be proved to be chaotic.
And it has some connection with problems of flames and turbulence.
So here's the trivia question for you Which three equations on this list won a Nobel Prize?
Anyone have a category candidate? Black Show.
Yes, Black. Charles Equation 1997 Economics.
Nobel Prize. MAXWELL No, because it was too early.
It certainly deserved a Nobel Prize, but it was already there when Nobel got going.
Schrodinger. Yes. So Schrodinger in 1933 won a Nobel Prize.
And the last. Another one that should have surely was Einstein's equations.
But Einstein famously, by best estimates, are that he deserved four Nobel Prizes.
He got one, and it wasn't for the Einstein equations.
So the other one is the Hodgkin Huxley equations for neural conduction.
And so that was the 1963 Nobel Prize in Physiology or Medicine.
So it's a wonderfully diverse here we have one in physics, Schrodinger, one in medicine, Hodgkin Huxley and one in economics.
And Black-Scholes. Interestingly, mathematicians take the view that, you know, linear problems are easy, nonlinear problems are significant.
There's even the expression linearity breeds contempt.
Nevertheless, if you look at the three that won Nobel Prizes, two of them are linear.
So Schrodinger is linear and Black-Scholes is linear.
It's only the Hodgkin Huxley equations which are nonlinear.
Okay.
I have another trivia that we won't go into here, but I believe that at least a third of physics Nobel Prizes have been won by eigenvalue problems.
So let's just glance for a moment at the PDT coffee table book on feet, just to give you a sense of that.
So the heat equation is the diffusion equation, and as the picture illustrates, it describes how edges turn into smooth things.
I hope you'll, in your own time read this. We try to write these things very carefully to say things that are interesting and important.
For example, there's the interesting mathematical fact that the solution is not unique.
It only becomes unique if you force it to be zero at infinity.
But then, more importantly physically is the wonderful, exciting connection between diffusion and probability.
Everything in our world that diffuses almost has some randomness as the underlying mechanism.
And because I like to demonstrate things on the computer, I'd like to play with that a bit.
I guess I'll write down a few things first, but then we're going to play with random effects that underline the underlie the heat equation.
But let's first treat it as a PD. So what I'll call Section 5.2 is our first numerical approach to solving PD e's by explicit.
Finite difference in one dimension. So when you see that phrase, you know that I'm talking about parties that have two independent variables,
namely one space variable and one time variable.
So the heat equation is the simplest such equation. Let's consider that one.
Suppose we have you t equals you.
X x the one d heat equation. And suppose we want to solve that numerically on the computer?
Well, the obvious thing to do is make a grid with space steps and time steps.
So let's do that. We'll have our grid of points in space.
And here we are at time zero. And then at the next time looks like this.
In the simplest case, we just have a regular grid in both space and time.
And the notation we often use is that K is often the timestamp.
So Delta T and H is often the space that.
So you can imagine this criticising the problem by approximating this space
derivatives by some finite difference in this direction and time in that direction.
Many ways to do this. For example. So the simplest formula.
I'll begin without even writing down the formula. I'll write down what's called the stencil for the formula.
So if you show the points that a formula connects in a diagram like this, that's called the stencil.
And what that picture means is I'm about to tell you an algebraic equation.
To get an estimate of this value, given estimates of these values,
the notation will use is I'll call this the end plus one j and that's this point here.
This point is then the end. J This would be V and Jerry plus one and V and Jane minus one.
So the subscript is telling you the x index and the superscript is telling you the time index.
So the simple formula you could use to model the heat equation would couple these four points.
And what we need to do is approximate the space derivative with these three and the time derivative with those two.
So it's pretty obvious how we should do that. We should say the end plus one minus the N divided by the time step.
Is equal to the j plus one minus to the J plus v j minus one.
Divided by the square of the space step.
So there you have the simplest possible finite difference formula for a p.
There are many things you can say about that formula. We're going to run it in a minute.
One thing you can say about it is that we could interpret it as a matrix of matrix multiplication.
So suppose I think of V.
As a vector. Of samples at different places in space.
So I could think of it as v one. V to.
Me capital, then I guess I'll call it.
So suppose I have capital n grid points in space, then the whole state at a given time is given by this vector of linked capital.
N and I could write this as a matrix problem.
So that stencil, that formula tells you this new value in terms of the other values at the previous time step.
So I could write it in this form.
I could say the vector of values at the new time step is equal to sum matrix times the vector values at the old times.
And that matrix has coefficients that come from this formula.
And in fact, it's a tri diagonal matrix. And it contains a constant on the diagonal and another constant on the first super diagonal.
And that same other constant on the first sub diagonal.
And A is equal to. One minus two.
K over H Square. B is equal to Kovrig Square.
So. Very generally, whenever you deal with finite difference approximations, at least linear ones, they can be formulated as matrix problems.
In this case, it's simply multiplying a vector by a matrix.
So you don't have to think of that as a matrix problem.
But in other cases, we'll be solving a system of equations if you want to do this on a computer.
Of course, if A is big, you'd be crazy to make this giant matrix as a matrix,
but you could use a sparse representation and just contain the numbers that matter.
The last thing to say about the way I formulated it here. I didn't talk about boundary conditions, but of course that's a big issue involving Pdes.
The simplest version of all of this would go as follows Imagine that I have an interval on which I want to solve the problem,
and suppose my boundary condition is that you is zero at both ends.
Then the natural thing to do would be to regard my unknowns as in the interior of that interval.
So this would be the one and this would be the end.
Of course, other boundary conditions would be treated in other ways, but for this case of zero boundary conditions.
What I've written down is exactly everything.
The boundary conditions are already there in the way I've written it, sort of from the fact that you'll notice the matrix of course,
doesn't contain a term there or there, but since that term is going to multiply zero anyway, it doesn't matter.
So this simple matrix form has implicitly imposed a couple of boundary conditions.
Let's run that thing. And this is the code called M35.
So if you look at M 35 heat, you'll see what it does is exactly this.
It uses a sparse matrix. So it actually does do it as matrix times vector just for fun.
And the matrix is spire so that that's not so inefficient. It actually constructs the main diagonal and then the off diagonal of the sparse matrix.
You can see the commands A times sparse and B time sparse.
There's the basic version of this code actually should have a percent sign a comment before the line that says comment is in for periodic B.S.
So the first version I'm going to show you is zero boundary conditions, as if that line had a percent sign in front of it.
So if I type 35 heat. You can see it flow.
A general principle of the heat equation is that it's exciting for a few milliseconds and then it grows boring.
Notice the boundary conditions are visible. You can see that it's zero at the end.
There's no unknown at the end. So the first point you see is the first in from the edge.
So there's the heat equation. Let's just run it once more. There you are.
It started with a non smooth initial condition, but infinitesimal amount of time is enough to make that smooth.
Notice that the space step I've used is 1/40, so a modest number of grid points and the time step I've used.
You can see the second line of code is 0.4 times x squared.
So I've taken a pretty small time step of 0.4 times x squared.
Let's run it again. But pause at each step.
So I have a variation called AM 35 pause.
And you can see I achieve that by commenting in the pause command.
So this is the initial condition. And then after one timestamp it looks like that mathematically this is now a smooth function.
Of course, computationally it still lives on a grid. And then very quickly you can see the shape changing.
It looks exactly flat in the top middle.
Mathematically, it's not exactly flat, but it's, if you like, exponentially close to being flat.
And then quickly you get. That behaviour.
Then there are two other things to modify with the comments. Suppose you wanted periodic boundary conditions.
So the value at this end, rather than being forced to be zero, is set equal to the value at this end.
Well, you can achieve that by making the matrix wrap around you.
Make it into a circular matrix in effect. There's a line of code that shows you that the one that breaks into the margin a little bit.
If I run that version, I get M 35 PR, so it looks the same and so far it is almost the same.
But now you can see at the boundary it's going up because now there's no heat being lost at the end.
The heat is just averaging out. So that's the one of the physical interpretations here.
We start with an uneven heat distribution. As time goes to infinity, it becomes a constant heat.
But the total amount of heat doesn't change with periodic boundary conditions.
Whereas for the initial one. With zero boundary conditions.
Again, it's the heat flow, but now the ends of the bar, if you like, are being held at temperature zero.
So he has flowing out and the destiny is zero temperature.
The last modification in this code to make is the one that.
Determines the size of the time step. So if you see in that second line of code, it's K equals 0.4 X squared.
And then the comment says, try changing point four, 2.5.
1.5 is the critical value at which the thing goes unstable.
So now let's change it. 2.51. So that's called n 35 u for unstable.
And you see this beautiful growth of instability.
Normally it doesn't look like that. That's what it looks like if you're right at the edge of instability.
Of course, normally you would have picked point seven or 3.2 and then you'd have explosion much faster.
So obviously there are issues to discuss concerning stability of these disparate transactions.
Now let's play with the other two codes on the page, which are the physical underpinnings of diffusion.
This is really one of the exciting stories of the last 150 years, if you like.
The the one minute version of the history is that you had these classical laws of continuum mechanics,
some of them invented here in Oxford, you know, Boyle's Law and so on, which in the 19th century people realised had a statistical basis.
So the two big names were Maxwell and Boltzmann,
who realised that it was random bouncing around of particles that were the basis of Boyle's law and other laws of continuum mechanics.
That was all a little bit abstract still in the 19th century, and then it became more concrete.
And famously with Einstein, another one of the Nobel Prizes, he didn't win once for his paper on Brownian motion in 1905,
when he really studied the details of how random effects could be visible on a macroscopic scale.
So he explained the motion of little particles in a fluid as the result of a lot
of impacts from even smaller particles in a fluid that were effectively random.
And then mathematically that became the basis of a greatly growing area of mathematics.
Norbert Wiener was a famous name who turned that idea of Brownian motion into a very precise
mathematical concept of a continuous curve that's nowhere smooth with a dimension of a half,
and all sorts of neat things going on, or one and a half, depending what you measure.
And in fact, in my career, among the two or three biggest developments in mathematics over the years that I've
been doing this is the growth of everything to do with probability and stochastic.
It's just amazing. So maybe 30 years ago you might have said the centrepiece of somewhat applied mathematics would be pdes.
Now many people would say it's probability. Personally, I still like beauty, and I think probability is a fad, but a very important fact.
So the word stochastic just gets more and more important in mathematics.
So let's go back to 1905. If Einstein had had MATLAB, this is what he would have done.
He would have run this code called M 36 Brownian.
And what it does is show you some Brownian motion. Now, Brownian motion as as the phrase is used by mathematicians,
describes a mathematical limit where things are really happening on infinitely small timescales.
So a Brownian path is a continuous path.
Of course, one simulates that by discrete things, and if you take a random walk, then that looks like a Brownian path.
And in a limit of more and more smaller and smaller steps, it can be proved it has the same behaviour.
So this code is taking random walks of a thousand particles.
So if I say M 36 Brownian.
You say a thousand particles start in the middle and then they're just moving independently.
Each one is moving around the block, but on a small enough scale that it's essentially Brownian motion.
The usual principle. At first it's interesting, and then it becomes boring.
So let's start it again. Pay attention. All sorts of things spring from this bit of physics.
This randomness is the source of the square root of t effect that you get in diffusion problems.
So what what's the radius of this blob?
Well, it scales like the square root of t all the square roots that appear and statistics are related to this picture.
It gets this universal square root effect that shows up in all kinds of ways and.
The more you study this stuff, the more addictive it becomes. You ask yourself, how big is how far out should the biggest one be?
That would be a question of extreme value statistics. And of course, all sorts of things are known.
What's the density function as a function of time? Well, that's easy.
That's a normal distribution and so on and so on.
In the limit of infinitely many particles, infinitely small time steps,
you get normal distributions and those arise in solving the heat equation and other diffusion equations.
You may notice the square effect, but that, of course, is due to the finite use of my screen.
The particles are not really confined to a square. There's another code that's even more addictive.
So if we had a spare hour, I would spend it all running the next code.
Let's run a little bit. This is called M 37 Rectangle.
Now, if I do in 37 rectangle.
Rectangle can. It ask me how many fleas do I want to put in a box of length?
Ten and with one. So let's begin with one flea.
So there's the flea. And when I press return, it wanders around.
And I don't know what's going on there, sir, until it hits the boundary.
So you can't resist wondering how long will that take?
Sorry about the weird graphics effects. See if I can prove that.
Let's give it more fleas. Oh, dear.
Let's try again, please. So each one keeps going until it hits the boundary and then it stops.
So of course, you can ask questions like. After a certain amount of time, how many do you expect to still be alive?
Let's take more fleas. But actually the way this problem arose for me was an interest in the question
of what's the probability of a flea reaching the end rather than a side?
So of course it might happen by chance that one of the fleas gets all the way to the end before hitting the side.
And you can see that that's going to be hard work. Let's take a thousand.
And in fact, this was one of the problems in the. Siam 100 to the challenge that I showed you.
I think the last problem was to determine the probability that this particle ends up at an end rather than at a side.
And the answer is something like ten to the minus six. And in fact, that probability can be determined exactly analytically for what it's worth.
You can use a conformal map of the rectangle onto a circle in order to find what would be called the harmonic measure of the end of the region.
Let's do it one more time and then we'll stop that.
With 10,000. Now, of course, in the world of gases, you have ten to the 23 fleas.
So, you know, a jar of gas would have roughly Avogadro's number of fleas in it.
Sorry about that. I have no idea.
Those of us on the campaign team have found that this year, the last 12 months, MATLAB seems to have gotten worse.
We hope this is a transient effect, but all sorts of weird things have been happening to us graphically,
and we're referring specifically to the graphics. We don't know how to fix them, but when new versions come out, things tend to be fixed.
So hopefully. Hopefully it will solve itself.
Okay. Let's kill that. Okay.
So you can see that probability, random walks, random effects are at the heart of important PD.
And to put that backwards, the pdes exist in order to model in a scientifically compact way, ultimately the effects of randomness.
But there are so many random particles in the physical world that although that may be the mechanism,
still the right way to do it is the continuous model for many purposes, of course, never for all purposes.
Now along those lines, I wanted to just make a remark or two about discrete versus continuous models.
So if you ask, you know, what's the truth? Is the truth discrete or continuous?
The more you think about that, the more you find that there are just too many things going on to give the single answer to that question.
And let me illustrate that by mentioning some things that are discrete and their analogues that are continuous.
So for example, molecules bouncing around are discrete, right?
And in some sense that's the truth of what's happening with continuum mechanics.
And then the continuous models that we use are of course continuous.
So continuum mechanics or the continuum models used in physics.
Our continuous and boil and hooked didn't know about all these molecules bouncing around.
All they knew was that they could measure the pressure of air and it would satisfy an equation.
Something else this discrete would be a finite difference.
Approximation. And that's a discrete approximation to a PD.
So in some sense that's the truth. But this is the mechanism we use for studying that truth.
Something discrete would be a random walk.
And when we say random walk, we we typically mean things like take a point and then go north, south, east or west with probability one.
And so you might follow a trajectory like this. So a random walk has some discrete time and space, maybe.
Or maybe just one or the other is discrete, but that's a approximation to a continuous object that mathematicians call Brownian motion.
But of course, Brownian motion to a mathematician, that's nothing continuous to a physicist.
Well, maybe that's something in the realm of bouncing molecules again.
So I really mean here. Brownian motion for mathematicians.
Another example since we've been showing pictures on the screen.
What about the dots on a computer screen? They are discrete at various levels.
They look discrete to you and they realise by pixels which are also discreet.
But then somehow we end up with perceiving things that are continuous.
So there are all sorts of complexities there.
Of course, your brain has its discrete rods and cones, business and discrete neurones, but somehow we end up perceiving it as a continuum.
You can see things are very mixed together. Let me just mention one more, which is floating point arithmetic.
Which the computer is using when I do all of these experiments.
But that's an approximation to real arithmetic.
And the list could go on. And what strikes me about a list like this is that the different rows really have nothing to do with one another.
The fact that this is discrete and that continuous completely unrelated to
this discrete versus that continuous and this discrete versus that continuous,
they're just all different. Things like this come up all the time in many different ways.
Another handout is an essay I wrote on this.
So the one with a pretty picture of Hurricane Katrina is an essay I wrote on discrete and continuous things.
Okay. I want to spend the last few minutes talking about something non-linear.
Our first example of a nonlinear PD, the Fisher CP equation.
Fourier was way back in 1807 or something, and he's the guy who did the heat equation, which is linear,
or at least he is a key person in that mostly the 19th century was about linear problems,
but a big counterexample is navier-stokes, which is a nonlinear equation, and that's deep into the 19th century.
In the 20th century, all sorts of good things happen in nonlinear problems.
So the Fisher CP equation is an equation that was independently studied by Fisher,
the great statistician in England, and then KP MP three Russian mathematicians.
So what? I even forget their names. What is it? That's cool.
McGrath, Piotrowski and his schooner. That's right. And Komarov is one of the greats of all time.
There's another page from the coffee table book that I handed out there.
So the point of this equation, physically, scientifically, is that the solutions of it usually are travelling waves.
If you're a mathematician, you can prove that other solutions also exist, but they tend to be unstable.
The stable solutions are travelling waves and they take a particular form.
Sorry, that looks like an infinite slope. I shouldn't have done that with a finite slope.
So at a particular time it will look like that and then it will move at a certain speed and indeed far away from boundaries.
There exist solutions with precisely this form of a constant shape that translate and translate at a constant speed.
So the equation is you t equals.
I'll put an epsilon in front of the diffusion term.
Plus you minus u squared. And this is prototypical of all sorts of pdes that blend together a linear bit of physics and a non-linear bit of physics.
So this this is the linear diffusion term.
And this is the nonlinear reaction term.
Diffusion is a reasonably clear word. Reaction sort of means pretty much anything nonlinear.
To model this, we can do just the sort of thing we did before.
The simplest model would be the end, plus one minus the N divided by K.
Is equal to. Well, you just do one term for that and another for this, and that will work.
So we would say, just as with the heat equation, we'd say v j plus one minus to the J plus v minus one.
Divided by H squared. So that's an H, so that's our diffusion term.
And then the reaction term we could put in simply like this v j minus v j square.
And that worked fine. Again, we're going to need a time stamp sufficiently small for it to work.
But assuming the time stamp is small, that works fine. And if we want we could write it in matrix form.
We would have V and plus one equals matrix times vector.
So that would be the linear part of the problem. But then, of course, we'd have to add in a nonlinear term, so we'd have to choose your own notation.
But it turns out it's the timestamp times this non-linear vector v j minus v square.
So I have a code that we can run for that. It's called M 38 feature KPP.
And you'll see it looks just like the heat equation.
Except that I've added this additional term k times you might achieve squared, but non-linear problems are always more interesting.
And let's see what it looks like. So this is M 38.
So that's the initial condition I've chosen.
And the reason for choosing this is that I want to illustrate this wonderful
feature of nonlinearity that you can have a preferred shape of a solution.
So this is just an arbitrary initial condition. But if I let time evolve.
Almost any initial condition will converge towards the shape of the travelling wave.
So look at this. And now it's exponentially close, if you like, to this way, which in principle would go forever except at the right boundary.
I have a boundary condition, but on an infinite domain that would simply go forever.
Let's run that a couple more times. So there's other initial effect which is complicated.
Not much travelling is happening yet, but then eventually it settles down and the slope gets a little less and it reaches this steady effect.
And of course, you can interpret this physically in all sorts of waves of heat or information flowing from one part to another.
This very simple model is philosophically, if you like, very much like the Hodgkin Huxley equations, which describe how pulses flow in neurones.
And the last thing I'll do is show you that it too goes unstable if you take the time, step too big.
So let's run the unstable version where I change .42.51.
So I say 30. Thank you for unstable same initial condition.
Who knows what's happening? Looks like a shark.
Okay. See you on Friday. I resigned.
Oh, yes.