The method of separation of variables shall first be demonstrated for a simple example. The method as described here will work as long as the spatial region is finite and has homogeneous boundary conditions.
The problem is to find the unsteady pressure field in a pipe with one end closed and the other open to the atmosphere:
We will try to find a solution of this problem in the form
There are two big reasons why the must be the eigenfunctions, rather than the :
(If the spatial range is infinite or semi-infinite, you may be able to use a Fourier transform. Alternatively, you may be able to use a Laplace transform in time.)
The first step is to find the eigenfunctions .
The eigenfunctions are found from requiring that each individual term of the form is capable of satisfying the homogeneous partial differential equation and the homogeneous boundary conditions.
In this particular example the partial differential equation is
homogeneous. But even if it is not, i.e. if the partial differential
equation was something like
By convention, is usually written as and
as in this step. To see when satisfies the
homogeneous partial differential equation, plug it in:
The trick is now to take the terms containing time to one side of the
equation and the terms containing to the other side.
While the right hand side, , does not depend on , you
would think that it would depend on the position ; both and
change when changes. But actually, does not
change with ; after all, if we change , it does nothing to ,
so the left hand side does not change. And since the right hand side
is the same, it too does not change. So the right hand side does not
depend on either or ; it must be a constant. By convention, we
call the constant :
If we also require to satisfy the same homogeneous boundary
conditions as . In this case, that means that at , its
-derivative is zero, and that at , itself is zero. So
we get the following problem for :
Note that the problem is completely homogeneous: satisfies both the partial differential equation and the boundary conditions. This is similar to the eigenvalue problem for vectors , which is certainly always true when . But for the eigenvalue problem, we are interested in nonzero vectors for which . That only occurs for special values of .
Similarly, we are interested only in nonzero solutions X(x) of the above ordinary differential equation and boundary conditions. Eigenvalue problems for functions such as the one above are called “Sturm-Liouville problems.” The biggest differences from matrix eigenvalue problems are:
Fortunately, the above ordinary differential equation is simple: it is a constant coefficient
one, so we write its characteristic polynomial:
We try to satisfy the boundary conditions:
Since we have a multiple root of the characteristic
equation, and the solution is
We try to satisfy the boundary conditions again:
solution of the ordinary differential equation is after cleanup:
We try to satisfy the first boundary condition:
We now try to also satisfy the second boundary condition:
The eigenvalues and eigenfunctions have been found. If we want to
evaluate them on a computer, we need a general formula for them. You
can check that it is:
If you look back to the beginning of the previous subsection, you may
wonder about the function . It satisfied
However, it is far more straightforward not to do so. Now that the
eigenfunctions have been found, the general expression for the
However, most people do solve for the corresponding to each eigenvalue . If you want to follow the crowd, please keep in mind the following:
Now that the eigenfunctions are known, the problem may be solved. To do so, everything needs to be written in terms of the eigenfunctions. And that means everything, including the partial differential equation and the initial conditions.
We first write our solution in terms of the eigenfunctions:
We know our eigenfunctions , but not yet our Fourier coefficients . In fact, the are what is still missing; if we know the , we can find the solution that we want by doing the sum above. On a computer probably, if we want to get high accuracy. Or just the first few terms by hand, if we accept some numerical error.
Next we write the complete partial differential equation,
, in terms of the eigenfunctions:
The above ordinary differential equations can be solved easily. For
each value of it is a constant coefficient equation. So you write
the characteristic equation
. That give
. Then the solution is
So, we have already found our pressure a bit more precisely:
To do so, we also write the initial condition and
in terms of the eigenfunctions:
Using the Fourier series for , , and above, the two initial
The Fourier series for becomes now
Now and are, supposedly, given functions, but how do we
find their Fourier coefficients? The answer is the following important
We are done! Or at least, we have done as much as we can do until someone tells us the actual functions and . If they do, we just do the integrals above to find all the and , (maybe analytically or on a computer), and then we can sum the expression for for any and that strikes our fancy.
Note that we did not have to do anything with the boundary conditions and . Since every eigenfunction satisfies them, the expression for above automatically also satisfies these homogeneous boundary conditions.