This note shows how to use the idea of eigenvalues and eigenfunctions to help guide finding a solution to a differential equation. There are many ways to solve this ODE, and this is a nicer more general way to looking at solving it.
Given
| (1) |
With some boundary conditions and
We start by rewriting this ODE as where
is an operator applied on
. This is just a
rewrite of the ODE, we did not do anything new here, but this way it makes the equation look more
like an
, and this helps, for later when we discretize it and apply FDM (finite difference
method), that what we will end up with. Also writing it as
is more cool, and makes one look
like a real math person.
Now that we have what to do? The whole point is to now find the eigenfunctions and
eigenvalues of the operator
(Recall, an operator has a matrix as a representation,
is a mapping
operator after all, so it is not far fetch to talk about an eigenvalues and eigenfunctions of an
operator.).
Let us now call the eigenfunctions of as
and the eigenvalues as
.
So now we can write
But how to find these ? For the above ODE, it is done by inspection as it is clear that
is an eigenfunction. We can see that because if we apply
to it, we
obtain
Hence it is now in the form
, where
, a scalar, and in this case
This is cool. We found the eigenfunctions and eigenvalues of Now what to do with them? Well,
Since from (1) we see that
is in the domain of the operator, because
, and we just
found the eigenfunctions of the operator, so then this is like saying that we found the basis vectors of
the domain of
where
lives, and we need to use the basis vectors of this domain to represent
. In other words,
| (2) |
The is just like in normal euclidean space, where we represent a vector as
The eigenfunctions are like the basis vectors
and
are like the coordinates of the
vector
. And
is like the vector
.
So far so good. We found the eigenfunctions of , and we rewrote
in terms of these
eigenfunctions. But wait a minute, we now have to find the
. These are like the coordinates of
when viewed in function space.
Here comes to the rescue something new that we need in order to make more progress. These eigenfunctions are not just some random things we pulled out of the sky. They are special functions and must adhere to some things. This is mathematics after all, and we must have some order.
These eigenfunctions must be orthogonal to each others and we define them on square integrable
space We just made this restriction of the space to be able to make more headway in
solving this problem.
What all this means, is that must be orthogonal to each others (just like
are
as a special case in the Euclidean space). Being in this space, we need to define an inner
product on them. We need to know how to perform an inner product between
and
.
You might feel tricked now, because we did not say any of this stuff about the eigenfunctions
when we found them above by inspection. But it is OK, luckily for us
does meet these requirements. How? because if we define the inner product between
and
using
Then the above becomes
So, are orthogonal to each others. This is what orthogonal means. If we inner product any
2 different eigenfunctions with each others, we get zero, but if we inner product an eigenfunction with
itself, we do not get zero.
Now we are really happy. We found that the eigenfunctions are orthogonal
to each others, and we can express
in term of them. We use this inner product
property to find
. We go back to (2) above, and multiply each side by
and
obtain
Integrating each side gives
But now we see that for
and zero for all other terms so the above reduces
to
Hence we just found
| (3) |
We take this and use it in (2). So, we have just found an expansion of
in terms of the
eigenfunctions
. i.e. we have found a complete representation of
as a function in the space of
, with its basis vectors and the coordinates
.
This is all so wonderful. But how does this help us to find the solution to well, if now
just write
, then we have
But hold a minute, this means that or
And this is the solution to the ode.
Hence given a differential operator , once we know its eigenfunctions and its eigenvalues, the
problem is solved.
We just have to express the forcing function in terms of the eigenfunctions, and once this is done,
the problem is solved. the solution is found. In real life, we obtain the matrix representation of ,
and we work on the matrix representation and find the eigenfunctions and eigenvalues.
So, solving this ODE becomes a problem of finding eigenvalues and eigenfunctions. But
remember, this all worked only because we were able to represent
in terms of the
eigenfunctions. If somehow we could not represent
this way, then this whole approach falls
apart.