Think of a drum and a drum stick. When you strike the drum it propagates waves and produces sound (acoustic waves). Now think of the same drum, but you are using several objects rather than a single stick to hammer the drum. The sound will be very different. However, you can use the idea of the stick to represent the other objects.

The idea is that you can learn from what that stick produced to analyze the larger impulses from other objects. This is exactly what the Green's function represents - the stick. Overall sound produced by any other objects can be represented via linear (or integral) combinations of the stick situation. There is a similar idea in signal processing. They don't call it Green's function - but it is referred to as impulse response function. Same idea. The Green's function works for continuous systems. The impulse response function works for discrete systems.

Thinking about only one-sentence-answer:

The Green function, of fundamental solution

(for the particular linear problem descrbed by PartialDiffEqns)

is the SOLUTION of this PDE,

but ONLY for the load applied at one point

(the point is understood in general way - can be a point in time or in space)

Using the Green function one can try further to construct the full solution of the problem

(e.g. for the full distributed load, satisfying all boundary conditions etc etc - this is somewhat mathematical adventure to do this for each particular problem)

In general, Green's function technique is used to deal with non homogeneous boundary value problem. In the derivation of the Green's function, the non-homogeneous part is dirac delta function which is physically means an impulsive force, Thus, Green;s function deals with the physical situation in the presence of an impulsive load.

Think of a drum and a drum stick. When you strike the drum it propagates waves and produces sound (acoustic waves). Now think of the same drum, but you are using several objects rather than a single stick to hammer the drum. The sound will be very different. However, you can use the idea of the stick to represent the other objects.

The idea is that you can learn from what that stick produced to analyze the larger impulses from other objects. This is exactly what the Green's function represents - the stick. Overall sound produced by any other objects can be represented via linear (or integral) combinations of the stick situation. There is a similar idea in signal processing. They don't call it Green's function - but it is referred to as impulse response function. Same idea. The Green's function works for continuous systems. The impulse response function works for discrete systems.

In Quantum Physics the Schrödinger Hamiltonian H can be regarded as an unbounded self-adjoint square matrix (often semi bounded from below). The Green's function (also called resolvent in the more mathematically oriented literature) is simply [H-E]^(-1), the inverse matrix of H-E, where E is any energy not belonging to the spectrum of H. The Green's function is extremely useful when one deals with "reasonable" perturbations of the so-called "free" Hamiltonian H0 since it is often possible to write the Green's function of the perturbed Hamiltonian in terms of the unperturbed one plus a compact operator.