Green's function

The typical wave equation, heat equation, Laplace equation are homogeneous equations of the form:

Lu (x) = 0, where L is a differential operator.

These equations can be written as:

If we want to find the solution u of the non-homogeneous ordinary or partial differential equation, it becomes

Lu (x) = f (x)

The concept of Green's function [1] through Poisson's equation below which is the non-homogeneous form of Laplace equation.

The non-homogeneous term f(r) on the right hand side could represent a heat source in a steady state problem or charge distribution in an electrostatic problem. 

Now, think of the source as the point source in which we are interested in the response of the system to this point source. If the point source is located at a point r', then the response to the point source could be felt at points r. The response is represented by the function G(r,r') called the Green's function. The response function would satisfy a point source equation of the form:

\begin{displaymath} \nabla^2 G({\bf r}, {\bf r}') = \delta({\bf r} - {\bf r}') \end{displaymath}

where, the term in the right hand side is the Dirac delta function. 

Solve the force oscillator problem using the Green's Function [1]