We’ve got one more section that we need to take care of
before we actually start solving partial differential equations. This will be a fairly short section that will
cover some of the basic terminology that we’ll need in the next section as we
introduce the method of separation of variables.
Let’s start off with the idea of an operator. An operator is really just a function that
takes a function as an argument instead of numbers as we’re used to dealing with
in functions. You already know of a
couple of operators even if you didn’t know that they were operators. Here are some examples of operators.
Or, if we plug in a function, say ,
into each of these we get,
These are all fairly simple examples of operators but the
derivative and integral are operators. A
more complicated operator would be the heat
operator. We get the heat operator
from a slight rewrite of the heat
equation without sources. The heat
operator is,
Now, what we really want to define here is not an operator
but instead a linear operator. A linear operator is any operator that
satisfies,
The heat operator is an example of a linear operator and
this is easy enough to show using the basic properties of the partial
derivative so let’s do that.
You might want to verify for yourself that the derivative
and integral operators we gave above are also linear operators. In fact, in the process of showing that the
heat operator is a linear operator we actually showed as well that the first
order and second order partial derivative operators are also linear.
The next term we need to define is a linear equation. A linear
equation is an equation in the form,
where L is a
linear operator and f is a known
function.
Here are some examples of linear partial differential
equations.
The first two from this list are of course the heat equation
and the wave equation. The third uses
the Laplacian and is usually
called Laplace’s Equation. We’ll actually be solving the 2D version of
Laplace’s Equation in a few sections.
The fourth equation was just made up to give a more complicated example.
Notice as well with the heat equation and the fourth example
above that the presence of the and do not prevent these from being linear
equations. The main issue that allows
these to be linear equations is the fact that the operator in each is linear.
Now just to be complete here are a couple of examples of
nonlinear partial differential equations.
We’ll leave it to you to verify that the operators in each
of these are not linear however the problem term in the first is the while in the second the product of the two
derivatives is the problem term.
Now, if we go back to (1) and
suppose that then we arrive at,


(2)

We call this a linear
homogeneous equation (recall that L
is a linear operator).
Notice that will always be a solution to a linear
homogeneous equation (go back to what it means to be linear and use with any two solutions and this is easy to
verify). We call the trivial
solution. In fact this is also a
really nice way of determining if an equation is homogeneous. If L
is a linear operator and we plug in into the equation and we get then we will know that the operator is
homogeneous.
We can also extend the ideas of linearity and homogeneous to
boundary conditions. If we go back to
the various boundary conditions we discussed for the heat equation for example
we can also classify them as linear and/or homogeneous.
The prescribed temperature boundary conditions,
are linear and will only be homogenous if and .
The prescribed heat flux boundary conditions,
are linear and will again only be homogeneous if and .
Next, the boundary conditions from Newton’s law of cooling,
are again linear and will only be homogenous if and .
The final set of boundary conditions that we looked at were
the periodic boundary conditions,
and these are both linear and homogeneous.
The final topic in this section is not really terminology
but is a restatement of a fact that we’ve seen several times in these notes
already.
Principle of Superposition
Now, as stated earlier we’ve seen this several times this
semester but we didn’t really do much with it.
However this is going to be a key idea when we actually get around to
solving partial differential equations.
Without this fact we would not be able to solve all but the most basic
of partial differential equations.