释义 |
Green's Function--Helmholtz Differential EquationThe inhomogeneous Helmholtz Differential Equation is
 | (1) |
where the Helmholtz operator is defined as . The Green's function is then defined by
 | (2) |
Define the basis functions as the solutions to the homogeneous Helmholtz Differential Equation
 | (3) |
The Green's function can then be expanded in terms of the s,
 | (4) |
and the Delta Function as
 | (5) |
Plugging (4) and (5) into (2) gives
 | (6) |
Using (3) gives
 | (7) |
 | (8) |
This equation must hold true for each , so
 | (9) |
 | (10) |
and (4) can be written
 | (11) |
The general solution to (1) is therefore
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 529-530, 1985. |