Differential equations in the sense of distributions[edit | edit source]
We can also generalize the notion of a differential equation.
The differential equation is a differential equation in the sense
of a distribution (i.e., in the weak sense) if and are distributions
and all the derivatives are interpreted in the weak sense.
Suppose is the generalized differential operator
where is infinitely differentiable.
We seek a such that
which is taken to mean that
Here is the formal adjoint of . We can check that
. If we say that is
formally self adjoint.
For example, if then
Therefore, for to be self adjoint,
In such a case, is called a Sturm-Liouville operator.
To solve the differential equation
we seek a distribution which satisfies
Define . Then must be a test function. We can
show that is a test function if and only if
Now let us pick two test functions and satisfying
Then we can write any arbitrary test function as a linear
combination of and plus a terms which has the form of :
which serves to define . Note that satisfies
Since , the action of on is given by
Therefore the solution is
where and .