Green's function ( Differential Equations) I am pretty fine on solving the partial differential equations using the method of separation of variables, now I am trying to understand the concept of Green's function for solving the PDE.And, I am really struggling with the concept of Green's function. It would be great if anyone could provide me any intuitive idea to understand it
 A: It is used to find a solution of an inhomogenous problem.
$$
L y = f
$$
In physics, it shows up as propagator, describing an interaction between two points in space due to a point source.
An easy example is the Green's function of the Laplace operator:
One looks for a function $G$ with
$$
\Delta_x G(x, x') = \delta(x-x')
$$
where $\delta$ is Dirac's delta distribution and
where $\Delta_x = \sum \partial^2/\partial^2 x_i$ is the Laplace operator acting on $x$ ($G$ is specific for a given differential operator and boundary conditions of the problem).
It would allow to create a particular solution
$$
y_p(x) = \int\limits_D G(x,x') f(x') dx'
$$
So $y_p$ at $x$ is the net result of the inhomogenity value at all other $x'$, mediated by $G(x,x')$.
$$
\Delta_x y_p = \int\limits_D \Delta_x G(x,x') f(x') dx'
=\int\limits_D \delta(x,x') f(x') dx'
=f(x)
$$
Here is the Green's function:
$$
G(x,x') = -\frac{1}{4\pi}\,\frac{1}{|x-x'|}
$$
For real world problems one has to consider boundary conditions on $G$ as well.
The full solution $y = y_h + y_p$ needs the solutions $y_h$ of the homogenous problem $L y = 0$.
