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