Applied linear operators and spectral methods/Differential equations of distributions

Differential equations in the sense of distributionsEdit

We can also generalize the notion of a differential equation.

Definition:

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

 

Note that

 

Therefore,

 

Here   is the formal adjoint of  . We can check that  . If   we say that   is formally self adjoint.

For example, if   then

 

Then

 

or,

 

Therefore, for   to be self adjoint,

 

Hence

 

In such a case,   is called a Sturm-Liouville operator.

ExampleEdit

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

 

and

 

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 equation (2).

Since  , the action of   on   is given by

 

Therefore the solution is

 

where   and  . Template:Lectures