Okay, so let's start off continuing from last week.
So, last week, what we did was we set up these iteration algorithms, right?
And the idea behind the iteration algorithms is to say.
If I have a differential equation, I can march forward in time, right?
Just by setting up some appropriate iteration scheme, based on Taylor
expansions. That's it.
And in the last lecture, we talk about is, not only do I know to setup the iteration
scheme, I can in fact decide what kind of error I want.
And also the stability of the scheme. Okay?
So we're going to put this into practice especially Wednesday.
But one other problem I want to talk about today is a boundary value problem.
Okay so these are fairly common. Let me write it down.
And, in some sense you already have an intuitive sense how to how to solve these
things, okay? So I'm going to go ahead and write
something down. This might be something like, some kind
of, let's write down like a second order differential equation.
And here's the deal with this. Unlike what we have before, where we said,
okay, if I solve this differential equasion, I specify the state of the
system, out of say, time zero, and I march forward into the future.
Here, it's a boundary value problem. So, I've still put t here.
T is still dummy variable. It could mean time, it could mean space,
it could mean anything, okay? And, really, what, what now the boundary
problem has, is something like this. You specify some end points.
So I'm going to solve this, differential equation.
From time, this in some interval, let say a to b, okay?
So I'm going to solve this thing from time a to b, and at time a, I impose this
condition, and at time b, I impose this. So for a second order differential
equation, to solve it, you have to impose two conditions.
For a first order differential equation, one condition.
For a third order, three conditions. For a fourth order, four conditions.
For a fifth order? Yeah.
You got the pattern. It's awesome.
Okay. It changes at seventeen where you have to
specify nineteen. I was just kidding.
>> [laugh] >> Okay, no so there you go whatever the order of the differential
equation to specify that number of conditions on it to nail down the solution
uniquely okay. You can still get a solution it's just
that you have an infinite number of them. So here when you specify that there's one
solution that comes out of this thing and you are trying to construct that.
>> Put the a in, in the middle. >> Typically what happens is you just
specify, the left. Like you mark it forward in time, or the
[inaudible] coordinates, that provided the two end points.
Yeah. That's normally the way people do that.
However, if you want to set something up you want to specify halfway across some
you could do that to. All right so let's, let's make this a
little simpler. And let's start off in fact here's the
generic form the most general form of the boundaries that typically written down.
I want to make something a little bit easier.
I'm going to start off again with second order different equation.
With this function f which is a general function like this.It's generically
non-linear let's say. But now let's make very simple boundary
conditions. So really, you're looking for some
solution, Some interval t and you specify that at time.
A, it's alpha, and it does something funny around here and then it ends up with, at
the other end with beta, okay? So in between, you have to satisfy this
equation, and at the boundaries you have to constrain it here.
So this is, this is, sort of the generic way to think about, doing that value
boundary problem, okay? Alright.
So that is the algorithm. Now let's try to tie this into what we
have been doing previously. So previously what we said is, the way
these iteration schemes worked, is I had to specify the state of the system, and
march it forward into the future. So you give me the state of the system at
some time, and from there I can march forward into the future, with like say
Runge-Kutta. I march it forward.
I specified everything to begin with. I have nothing left.
Normally, in other words, I cannot take off on Runge-Kutta unless I specify
everything at the beginning. And this says well, I haven't specified
everything at the beginning. Some of it I've specified here, some over
there. So how do I make sense of using the
techniques we've already developed with this thing here, okay?
Cuz this thing here is saying, well I'm specifying things here and into the
future. It's a little bit like I'm specifying one
thing. I start marching forward, and I'm supposed
to hit a target. And that's exactly kind of what, this is
the way we're going to use this routine.