## Green's Function

Let

 (1)

be a differential Operator in 1-D, 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 3-D, the Green's function is defined by analogy with the 1-D 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 Equation--Green's Function,'' Green's Functions--One Dimension,'' and Green's Functions--Two and Three Dimensions.'' §8.7 and §16.5-16.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 480-491 and 897-924, 1985.