Greens Functions, PDEs and Laplace Transforms

[John Creighto]

According to wikipedia the greens function is defined as:

$$L G(x,s) = - \delta(x-s)\,$$
http://en.wikipedia.org/wiki/Green's_function#Definition_and_uses

when L is a differential equation then the greens function is the impulse response of the differential equation.

If a Hilbert space can be found for the operator then the greens function is given as follows:

$$K(x,y)=\sum_n \frac{\psi_n^*(x) \psi_n(y)} {\omega_n}$$
http://en.wikipedia.org/wiki/Fredholm_theory#Homogeneous_equations

Where $$\phi$$ are the eigen vectors and $$\omega_n$$ are the eigenvalues of the operator. (Not sure how unbounded basis are dealt with).

For ODEs we can find the eigenvalues by finding the poles of the Laplace transform. I'm wondering if there is some generalization of the Laplace transform for partial differential equations. The form of the resolvent:

$$R(z;A)= (A-zI)^{-1}.\,$$

http://en.wikipedia.org/wiki/Resolvent_formalism

Looks strangely similar to part of the solution when solving for S the Laplace transform of a system of first order linear differential equations. Also with regards to generalizing with respect to partial differential equations, I presume a convolution with a greens function turns into a multiple convolution over several variables.

Thinking in terms of ODEs the poles of the resultant should be the eigenvalues.

 

[John Creighto]

Here are some relevant links:
http://en.wikipedia.org/wiki/Fundamental_solution

Laplace Transforms and
Systems of Partial Differential Equations[/url]
A. Aghili and B. Salkhordeh Moghaddam

transform pairs of N-dimensions
and second order linear partial differential
equations with constant coefficients[/url]
A. Aghili, B. Salkhordeh Moghaddam

So it appears that there are multi-dimensional versions of the Laplace transform that can be used to solve Partial Differential equations. Any incite anyone has on this would be greatly appreciated.