# Introduction to Laplace and Poisson Equations

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
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