Throughout this chapter we’ve been talking about and solving
partial differential equations using the method of separation of
variables. However, the one thing that
we’ve not really done is completely work an example from start to finish
showing each and every step.
Each partial differential equation that we solved made use
somewhere of the fact that we’d done at least part of the problem in another
section and so it makes some sense to have a quick summary of the method here.
Also note that each of the partial differential equations
only involved two variables. The method
can often be extended out to more than two variables, but the work in those
problems can be quite involved and so we didn’t cover any of that here.
So with all of that out of the way here is a quick summary
of the method of separation of variables for partial differential equations in
two variables.
- Verify that the partial
differential equation is linear and homogeneous.
- Verify that the boundary
conditions are in proper form. Note
that this will often depend on what is in the problem. So,
- If you have initial
conditions verify that all the boundary conditions are linear and
homogeneous.
- If there are no initial
conditions (such as Laplace’s equation) the verify that all but one of
the boundary conditions are linear and homogeneous.
- In some cases (such as we
saw with Laplace’s equation on a disk) a boundary condition will take the
form of requiring that the solution stay finite and in these cases we just
need to make sure the boundary condition is met.
- Assume that solutions will
be a product of two functions each a function in only one of the variables
in the problem. This is called a
product solution.
- Plug the product solution
into the partial differential equation, separate variables and introduce a
separation constant. This will
produce two ordinary differential equations.
- Plug the product solution
into the homogeneous boundary conditions.
Note that often it will be better to do this prior to doing the
differential equation so we can use these to help us chose the separation
constant.
- One of the ordinary
differential equations will be a boundary value problem. Solve this to determine the eigenvalues
and eigenfunctions for the problem.
Note that this is often very difficult to do and in some cases it will be
impossible to completely find all eigenvalues and eigenfunctions for the
problem. These cases can be dealt
with to get at least an approximation of the solution, but that is beyond
the scope of this quick introduction.
- Solve the second ordinary
differential equation using any remaining homogeneous boundary conditions
to simplify the solution if possible.
- Use the Principle of
Superposition and the product solutions to write down a solution to the
partial differential equation that will satisfy the partial differential
equation and homogeneous boundary conditions.
- Apply the remaining
conditions (these may be initial condition(s) or a single nonhomogeneous
boundary condition) and use the orthogonality of the eigenfunctions to
find the coefficients.
Note that in all of our examples the eigenfunctions were sines and/or
cosines however they won’t always be sines and cosines. If the boundary value problem is
sufficiently nice (and that’s beyond the scope of this quick introduction
to the method) we can always guarantee that the eigenfunctions will be
orthogonal regardless of what they are.