Last time we took our first step toward PDEs

from ODEs

PDEs more complicated

describe continuous effects effects from continuum

Sciences where you have a distribution of mass where you have instead of having discrete objects you now have

continuous objects like a guitar string like a drum or

Like which will be a very key example for today the temperature distribution in this room

So we talked a little bit about the wave equation

because one of the more poetic points that I made was that the second derivative is responsible for Movement and

therefore life and I like that point and so we talked about

The wave equation which is all about movement and oscillations

But right now. We'll take a little bit of a step [back]

And I'll just give you a little bit of an overview of the kinds of ODEs

Excuse me of the kinds of PDes that people solve most commonly, so that's the title of today's lecture, and I [missed] I

skipped the most important word that

characterizes all of these

odie's that excuse me PDEs

I've got to change shift gears to PDEs all of these most fundamental PDEs that we'll talk about

So I just want to give you an idea of the physical problems that give rise to these equations

This so what they're called the sorts of solutions that we might expect

The certain approach is that we might take to find those solutions?

The common boundary conditions that we would consider in in some cases initial conditions

The simple [analogous] that you can think of from linear, Algebra and ordinary differential equations this time

I did not misspeak because some of these are in some ways analogous to ordinary differential equations, and that will give us

Expectations for what we're going to find

But what all of these equations have in common is

Characterized by one word that all of you know

obviously

That will go right here. Which is probably the most important care to ristic of these equations, so

Anybody wants to guess this word one more time. We had alternatives

continuous totally boring

Conic because these are conic sections yes, yeah, no

not conic more fundamental

Who said that hey linear? I'll put it in when we get there. Maybe they're linear. Maybe they're not

So keep that keep this word in the back of your minds are these linear

what does it mean in what sense and why would it be helpful [and]

Are they are they they are but maybe not they are?

Okay, so we started by introducing the laplace operator or the laplacian

Some really good words today. [I] mean good names

Laplacian [I] should write it down

it's another guy who's a privilege to speak about a

very distinguished French Mathematician

Work who worked for Napoleon [does] that sound a little funny worked for Napoleon?

Makes you long for the times where success was correlated with merit?

So this is called the laplacian and we know its main property

It meant not so much main property, but it's invariant property something that has nothing to do with coordinates

It is a measure if you take a function of two variables and evaluate its laplacian at a point

It gives you a measure of how different the value at that point is

from the average of its neighbors and

We know that there are certain functions

That we think of in the equilibrium state as being

Totally averaged out and the primary example that you should always think of when you hear

This is the temperature distribution

Let's take the temperature distribution in this room

here we would have three variables xy and [z] because we live in the [three-dimensional] space which you can imagine may be a

[two-dimensional] space being good enough analogy and

You would all agree with me that if it's cold there

And warm there than here, it'll be average of those to some not average, but somewhere in between

there cannot be a spike in temperatures just something it's just not anything that we've ever experienced if

You were to light a candle [now]

There is a spike in temperature right where the candle is burning if you if you put the candle out it

will very quickly relax back to the equilibrium and

Everywhere the temperature will be more will be the average of the neighboring values

And you might say well, that's not exactly true. Look at me

I'm a I'm at a hundred degrees Fahrenheit then the room around me is at 70 degrees Fahrenheit. Well you that's right

There are processes inside you that

maintain that temperature

You're burning fuel so it's equivalent to burning a [candle] if that process God forbid was extinguished

You would quite quickly or maybe relatively slowly depending on your material properties

average out

So the temperature in the room is

characterized by the fact that it's laplacian equals zero by zero

Laplacian and

It's this equation, so if you imagine that this is your [2-dimensional] room

the temp the equilibrium temperature distribution

Once it's settled and not changing anymore is described by this

Laplace's equation. This is called laplace's equation when you have the laplacian

equals zero, we'll have a lot more to say about this, and that's what we will solve a

Few more things I [want] to say about this so the equilibrium temperature distribution

You must always say equilibrium the equilibrium temperature distribution is the quintessential

field characterized by zero laplacian

There is another good word that I'll mention right now

functions whose laplacian is zero in other words who are at every point the average of its neighbors are called Harmonic a

Function is harmonic if [it's] laplacian is zero

harmonic

But I have a different [example] in mind when I think of harmonic functions. [I] have an example. That's actually wrong

But I find it more visual, so I'll tell you [about] it. So actually imagine something in two dimensions

so now we only have x and [y] and

So the function itself you can be imagined along the z axis and so the picture would be something like this

and

This domain is right here. It will look flatter because we're looking at it from a different angle, but it's the same domain and

So this is our domain of definition, and you're once again beginning to see some of the complexity associated with

partial differential equations it all happens on a domain and

So describing the domain to begin with is something that you need to do before you [can] solve the [equation], so that's complexity right there

Okay, and so the function you

Would live with you can actually graph it

Because we have three things x y and the function u it's nice [that] you can graph it the temperature in this room

You cannot graph it you have to color code it because we're run out of dimensions

So I would imagine it on the boundary. Maybe having values like this

All right, so imagine [that] these points all project down to the boundary does that make [sense], okay?

[so] we will discover in a moment that this is the kind of equation for which you need to set boundaries conditions

In other words to even begin to attempt solving it you need to know you need to be given not just this differential condition

But the values of the function on the boundary, we'll get back to temperature

And you'll see that that's kind of what we need to do

Okay, [and] here's what I imagine even [though] the system that I what that I'm about to describe is actually described by a different equation

much more

Beautiful in some ways imagine that this is a wire

frame that you dipped in a Soapy solution, and you pulled it out, and you imagine the soap film that forms and

You know very well that soap film does not have any bumps are they in an expected bumps. It's as smooth as possible

Some people say that. It's as flat as possible

So that's the picture that I have in mind

Actually this surface

This shape is described not by this equation

But by something much more complicated. It says that mean curvature when you study differential geometry

equals zero

But it just corresponds very nicely to my intuition. What harmonic functions. Don't do is

Go smoothly like this, and then have a little maximum

Little local maximum and then continue they just don't have those bumps. They're completely smoothed out

You couldn't average it out

Anymore that's what harmonic equations look like

Excuse me

That's what harmonic functions look like and that's my intuition [this] soap film is my intuition

for what I expect out of a

[out] of laplace's equation

Now let's talk about boundary conditions a little bit to find the temperature distribution in this room

There are certain things that you will want [to] know

For instance which are met equilibrium temperature [distribution] for instance

Yes, we are here burning fuel right that certainly factors into the equilibrium temperature of the room

So if you had to mathematically not with a thermometer, which is the most direct way of doing it?

But by solving a partial differential equation

Find the temperature distribution in this room. You would not what you would want to know where the [people] are

Because wherever I am that's where the temperature is a hundred degrees

[98] Point I'm I don't have a fever don't worry

So whatever it should be that's the temperature that it is if I put my arm right here and wait for the temperature to equilibrate

Equilibrated sounds right then the temperature right here will be a hundred degrees I

Impose that condition by putting my arm there and if you want to do some big calculation you want to know where [my] arm is

So basically you want to know all the surfaces

Where the temperature is given?

So and that's usually happens on the boundary

So you have to be given the values of the temperature and the boundaries [or] else you don't have enough information to

Solve the system there are very many fields that are harmonic in this room

But but only one that has prescribed temperature on all of the boundaries

So you have to know temperature on the board and so forth?

Everywhere floor ceiling. I don't know why singled out the board there is one alternative

There might be something like a window where you know that the heat is escaping

So if you know the rate at which the heat is [escaping] across a boundary. That's also a sufficient condition

Now you might say well, it's getting cooler. If heat is is

escaping well not really because heat is also coming from the radiator and there can be a balance and

So on every boundary you need to know either what the temperature [is] or what the rate of heat flow is

But you need to put the entire bounder one way or the [other]

You need to know what goes on and that when those are called boundary conditions

and I just want to tell you right now that if I were to somehow describe the shape and

tell you in whatever way I can whether it's by equation or just tabulating the values or

parameterizing the curve and giving you a function of that parameter if I somehow prescribed the temperature on the Boundary and

Told you to find the function that satisfies this condition

Given this differential condition given the boundary condition you would have no idea how to approach this problem

It will actually get a little bit easier because this is a simultaneous

balance of a lot of things

It's not like an od where you can start somewhere, and then start rolling it. This is a very different thing. This is a simultaneous

satisfaction of

Conditions like this at every point. So you have to somehow so for all points all at the same time

It's not at all [like] [an] od

so it's very different, and it's a little bit surprising that will actually start right here, and

Then you will again running ahead a little bit you might get a little bit suspicious

Because I've always said

for ODEs that the order of the equation

matches the number of conditions that you need and

This is a second-order equation right there is [their] second derivative?

I'm just looking for an analogy

So you'll see that the analogy maybe doesn't work sometimes analogies work sometimes they don't so in this case

I just pulled out an analogy that doesn't work well that's interesting in its own way, but it kind of will work

So it's a second-order equation. We're dealing with second derivatives in x and also with second derivatives in y

Yep, there is only one condition [I]

[don't] specify a pair of conditions. What the function is and what its derivative is

So is that a mismatch?

well we find out that's not a mismatch mismatch is actually just perfect because it kind of is [like] two conditions if you kind of

mentally

Divide this into two halves. It's like there is a condition on the left and there is a condition on the right

So that will end up being [not] a bad way of thinking about at least

reconciling the order of the equation with the number of conditions there are two sides always opposite sides and I'm

Specifying a condition on the left and on the right I have to get all the boundaries

the same thing would work with an od if instead of initial position and initial Velocity I

specified an initial position and a final position

From whatever point of view you look at odie's that would work

You know because you'll have your C one times one thing plus C two times another thing

So you need to determine C [one] and [C] [two] and there you have two conditions?

You don't really care what those two conditions are so there is an analogy

Okay, that's all I want to say about laplace's equation at this point. I will say one other thing

There is a very much a related equation

[in] this lecture, we'll just keep marching there are several things

I have to say about each one of these equations

So I'll just say them one at a time about each of the systems. We'll come back to laplace's equation, but there's also parsons equation

Poisson's equation

Which says laplacian of U is

not zero but some given function f ah

What's an analogy? I [think] an analogy would be a burning candle and

so there is

A source, a reason why the function is not harmonic.

Obviously, it's hotter where the candle is and colder everywhere around. That's a maximum

It's very much not the average of its neighbors

So [when] there is a reason when there is a source then when there's an influx like that then the harmonicity

of the function might break and so you would deal [with] a function like this. They're [very] closely related