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

 

or,

 

Therefore,

 

Now,

 

Hence,

 

And,

 

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

 

with

 

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

 

ExampleEdit

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.

Therefore,

 

For the heat equation

 

we can take

 

That gives us   and we recover the same   as before.


In the general case, the solution is

 

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