Applied linear operators and spectral methods/Differential equations of distributions

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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

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.

Example[edit | edit source]

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