Applied linear operators and spectral methods/Greens functions 2

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Green's functions for linear differential operators[edit | edit source]

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

Example[edit | edit source]

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