but now I want to talk about this most important word that's missing from here which was

linear linear

Are they linear? What would it mean for them to be linear and

How would that help us to solve them?

Well, I'll I won't hold you in suspense. I can tell that the suspense is just killing you

So yes, they are linear and here is the simple thing that it means

If I imagine that there are two two different functions that satisfy this equation

two different functions

That satisfy this equation two different harmonic functions, and I add them together

Will they still satisfy the same equation?

Yes, I don't. I don't even want to discuss it because it's so obvious

discussing it will make it seem complicated, but you look here and

If you have two functions, and you take their second derivatives and then add together the second derivatives here. I am discussing it

It's like adding the functions first and then taking the second derivative so the second derivative always factors out of course

They're linear of course it is a linear function

So I always make this discussion a little bit incomplete because you should also ask the question if they have one function that satisfies this

Equation would a scalar multiple of that function and number times that function?

Also satisfy this equation and the answer is just as clearly yes, I see lots of thumbs up

So yes, this is a linear equation it

Is also now that it's Linear I?

Can now start using a lot of the linear Algebra terminology? This is a question of the null space isn't it?

That says find a function in the null space of the laplace operator

Okay, that makes us more comfortable. I think just

It's almost like Ax equals zero

it has that feel of ax equals zero and

If that and if this is ax equals zero, that's the linear Algebra analog

Then what's the linear Algebra analog of this?

Ax equals B. So I don't have too much space to write it

But I think it deserves to be written down ax equals zero here. That's laplace's equation

Ax Equals B

here

There you go, and so now you kind of know if you were

To start let me back up half a half a step and say this gives us a little bit of hope in

Thinking about how we would solve this these problems

Because we can start in a way building our solutions up

From the many many functions that have this property

Because the two functions satisfy this equation then there's some satisfies this equation

So you start thinking about solving these problems by decomposition you?

Will say I will find one function that satisfies this equation

Not so hard

X squared minus y squared would you agree?

Equals zero the laplacian of the function equals zero that's just one example, but of course there are more than infinitely many

2xy x cubed minus 3x squared y

So easy to think of take any one of them and shift them I?

Can think of a combination with sines and cosines?

Just so that they you get a minus sign in one and none at the other and they totally cancel each other

perfectly I can think of a lot of things and

My strategy would be to just find as many of these as I can

They all satisfy this equation, but remember that

The hard part is the boundary condition

And then I'll start putting them together in just the right combinations

to Satisfy boundary conditions and as long as I deal only with harmonic function and

as long as I deal with harmonic functions only I don't have to worry about this being satisfied this becomes the easy part the

Hard part is maxing the boundary condition

So that's basically what?

Just about the only thing that you can do analytically with these equations

And you might say but keep wait wait wait. You said linear is easy Nonlinear. It's hard well

Yes, that's true. But with pDes it becomes only partially true

No pun intended because you have this complicated domain

Which is not square?

You know that's what makes it hard

Unavoidably hard a Pde generally speaking cannot be easy. You cannot squint at it and know the answer. It's work

Because there's this complicated shape and why is there a complicated cable because life has complicated shapes

And we don't care about problems if they don't come from well. We do we're mathematicians, but we always remember where they came from

Or else. What's the point now? This one is not a linear equation

You guys agree with me

Because if you have two solutions that satisfy this equation, and you add them together their laplacian will of course be 2f

But here that other analogy with Odie's well, actually it's not an analogy with it

It's exactly which you were saying this will be something like a particular solution that gives you f and

Then anything from the null space any harmonic function, and that's how you would probably approach solving this problem

you won't even pay attention to boundary conditions you will just find one function whose laplace seen equals f somehow and

then you will start at

Adding as harmonic functions in just the right combinations to match the boundary condition

and you're sure given that f will change them a little bit given that the

Particular Solution will have to adjust them a little bit

but if we approach the strategy if we use the strategy we don't have to worry about the

The part that you might think is hard when you just start looking at these which is this part now?

That's not the hard part matching the boundary conditions is the hard part

Okay, this is a linear equation also

there's everything linear on the right there's

everything's linear on the left

So this will solve this same way also

We'll find one possible function that satisfy this another 1/3 1/4 infinitely many

And then we'll and then we can add them together all we want in this condition is still satisfied

we just got to get the initial conditions and the boundary conditions and

not worry about this once again solution by decomposition

Same thing here. No change by this fundamental notion that we can find as many functions

That is one that satisfy these conditions in an aTom

And when I say as I'm I really means combine them in linear combinations also multiply by numbers

All of that that this basic idea applies to all of these functions all of them

all of them so all of these equations if you want to solve them analytically

You usually resort to solution by decomposition

Such an important concept

first week of Linear, Algebra

second week and

That's and that's what we're still doing here today

solution by decomposition, so the most important word is

Linear okay, so what are these little things?

So these linear equations have names associated with them which I don't like because it's much better to say describe to the equilibrium

Describe slow Convergence to the equilibrium

Describes the rush to the equilibrium with overshoot

That's how I prefer to describe them, but they're actually called Elliptic Parabolic Hyperbolic

Elliptic you know that I wrote down the equations for the circles

but an ellipse is not that much difference kind of like the sum of square is equal something and

Here we have the sum of second order derivative, so it's almost like x squared plus y squared

That's why it's a it's a metaphor. Not a metaphor

But an analogy just to find a very nice name for an equation like this, so that's why these is called elliptic

These is called parabolic because it's almost like something equals something squared, right?

This is the sum of Squares, and this is something so it's almost like y equals x squared

in some analogy, so these are called parabolic and

these are of course something like a

Square on the right a square on the list instead of adding together squares

We're subtracting squared

and so these are called hyperbolic because that's the equation of a hyperbola and

Now I'll sort of remind you of an of another point that I made you how one sign

Can make all the difference in the world in PDEs so a little bit of a formal I'll just make

Sort of a formal point

so this is laplace's equation right it's

It's elliptic

describes a Snapshot in Time I

Maintain that it is hard to solve

Because you don't know where to start you don't have any where to start you have to do it all simultaneously?

Right and the moment you do this

It becomes a wave equation

Describes the rush to the equilibrium with overshoot and so it's not surprising that they're a completely different

problem

When you study PDEs you your specialization is much more narrow than this you wouldn't say I'm an expert in

Hyperbolic equations, that's too broad

You're an expert in a much finer subset of a particular kind of equation

So that that's the world of PDEs and it's not mysterious

Why a minus sign makes all the difference in the world? How you approach the problem

How you think about the problem the sorts of things you'll find?

ok so these are the three fundamental types of linear PDEs and

So I think we'll be spending the bulk of our time with these equations