Distribution theory and differential equations.
Below is the minimal mathematics - or rather the minimal
Linear Systems Theory
- needed to understand the role of distributions and Green's functions with
differential equations. As extracted from old notes about "Signals and Systems".
Signals are represented by $\,x(t)\,$ and $\,y(t)\,$ where $\,x\,$ is an input
signal / excitation, $\,y\,$ is an output signal / response and $\,t\,$ is time.
A linear system $S$ is represented by $y(t) = S x(t)$ ; it
produces an output when given an input and linearity means that:
$$
S \left[ \lambda\, a(t) + \mu\, b(t) \right] =
\lambda \, S \, a(t) \, + \, \mu \, S \, b(t)
$$
So $S$ is a linear one-dimensional operator. More about linear operators
and especially operators with (ordinary) differential equations in:
A system $S$ is homogeneous in time - also called
[Time invariant](http://en.wikipedia.org/wiki/Time-invariant_system) or
[Shift invariant](http://en.wikipedia.org/wiki/Shift-invariant_system) - iff for all input signals $x(t)$
and for all output signals $y(t)$ and all time-shifts $\tau$ :
$$
S \, x(t-\tau) = y(t-\tau)
$$
Properties of linear homegeneous systems are for example:
$$
S x'(t) = y'(t)
\quad \Longleftrightarrow \quad
S \frac{d}{dt} x(t) = \frac{d}{dt} S x(t)
\quad \Longleftrightarrow \quad
\left[ S \, , \, i\, \hbar \frac{d}{dt} \right] = 0
$$
The response of the derivative of the input is the derivative of the output;
time differentiation cummutes with the operator of the system; conservation
of energy is guaranteed (QM).
Consider the following summation, for a broad class of functions $f(t)$ :
$$
S \left[ \sum_i f(\tau_i)\, x(t-\tau_i)\, \Delta\tau_i \right] =
\sum_i f(\tau_i) \, S x(t-\tau_i) \, \Delta\tau_i =
\sum_i f(\tau_i)\, y(t-\tau_i)\, \Delta\tau_i
$$
Taking the limit of this Riemann sum for $\Delta\tau_i \to 0$ yields:
$$
S \left[ \int_{-\infty}^{+\infty} f(\tau)\, x(t-\tau) d\tau \right] =
\int_{-\infty}^{+\infty} f(\tau)\, y(t-\tau) d\tau
$$
Where the convolution integrals $f * x$ and $f * y$ are recognized.
Suppose that a linear and homogeneous system is excited with a Dirac-delta as
its input. Then the corresponding response is called a delta response,
written as $\,h(t)\,$ by definition. So we have:
$$
S\, x(t) = y(t) \qquad ; \qquad S\, \delta(t) = h(t)
$$
This one-dimensional function is equivalent to Green's function when
generalized to more dimensions e.g. space-time.
The fundamental property of Dirac-delta says that: $x(t) = x(t) * \delta(t)$ ,
called "diafragma" property in Dutch, but couldn't find a nice English equivalent.
Hence:
$$
y(t) = S\, x(t) = S \left\{ \, x(t) * \delta(t) \, \right\} = x(t) * h(t)
$$
Thus the superposition integral of $S$ has been found:
$$
y(t) = h(t) * x(t) = \int_{-\infty}^{+\infty} h(\tau)\, x(t-\tau) d\tau
$$
Consequently: if we know the Delta-response then we know any response of the system.
The above explains in a nutshell some essentials, at hand of one-dimensional
linear & homogenous systems in time. I hope it nevertheless serves a purpose and
that the reader is capable of thinking how to generalize this material to more than one dimension.