Green's function ( Differential Equations)

Question

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

Answer 1

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 $L y = 0$.