Applied linear operators and spectral methods/Greens functions 2

Green's functions for linear differential operatorsEdit

Consider the one-dimensional heat equation given by


This equation has Green's function   which satisfies


The Green's function is given by












Note that the second derivative of   is a delta function.

We can use this observation to arrive at a more general description of the Green's function for a particular differential equation. That is, for a  th order linear differential operator we would want the  th derivative of   to be like a delta function. Thus the  th derivative of   should be like a Heaviside function and all lower derivatives should be continuous.

In particular, consider the operator   acting on   such that




where  . If we integrate this equation across the point   from   to   we get


This condition is called a Jump condition.

This suggests that the Green's function   satisfying


in the sense of distributions has the properties that

  1.   for all  .
  2.   is continuous at   for  .
  3.  .
  4.   must satisfy all appropriate homogeneous boundary conditions.

If the Green's function exists then



Let us consider a second order differentiable linear operator, i.e.,   on   with separated boundary conditions,


where   and   are boundary operators.

The differential equation


and its required continuity conditions are satisfied is


The first two terms above are to satisfy the boundary conditions while the third terms gives us continuity.

The jump condition is that


where   is the Wronskian.



For the heat equation


we can take


That gives us   and we recover the same   as before.

In the general case, the solution is