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.


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




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 equation (2).

Since  , the action of   on   is given by


Therefore the solution is


where   and  . Template:Lectures