Let

(1) 
be a differential Operator in 1D, with Continuous for , 1, ...,
on the interval , and assume we wish to find the solution to the equation

(2) 
where is a given Continuous on . To solve equation (2), we look for a function
such that
, where

(3) 
This is a Convolution equation of the form

(4) 
so the solution is

(5) 
where the function is called the Green's function for on .
Now, note that if we take
, then

(6) 
so the Green's function can be defined by

(7) 
However, the Green's function can be uniquely determined only if some initial or boundary conditions are given.
For an arbitrary linear differential operator in 3D, the Green's function
is defined by
analogy with the 1D case by

(8) 
The solution to
is then

(9) 
Explicit expressions for
can often be found in terms of a basis of given eigenfunctions
by expanding
the Green's function

(10) 
and Delta Function,

(11) 
Multiplying both sides by
and integrating over space,

(12) 

(13) 
so

(14) 
By plugging in the differential operator, solving for the s, and substituting
into , the original nonhomogeneous equation then can be solved.
References
Arfken, G. ``Nonhomogeneous EquationGreen's Function,'' ``Green's FunctionsOne Dimension,'' and
``Green's FunctionsTwo and Three Dimensions.''
§8.7 and §16.516.6 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 480491 and 897924, 1985.
© 19969 Eric W. Weisstein
19990525