PDES: SEPARATION OF VARIABLES AND OTHER METHODS
for example, the electrostatic potentialu(r) due to some distribution of electric
chargeρ(r). The electrostatic potential satisfies
∇^2 u(r)=−ρ
0,whereu(r)→0as|r|→∞. Since the boundary condition on the surface at
infinity is homogeneous the surface integral in (21.86) vanishes, and using (21.90)
we recover the familiar solution
u(r 0 )=∫
ρ(r)
4 π 0 |r−r 0 |dV(r), (21.91)where the volume integral is over all space.
We can develop an analogous theory in two dimensions. As before the funda-mental solution satisfies
∇^2 F(r,r 0 )=δ(r−r 0 ), (21.92)whereδ(r−r 0 ) is now the two-dimensional delta function. Following an analogous
method to that used in the previous example, we find the fundamental solution
in two dimensions to be given by
F(r,r 0 )=1
2 πln|r−r 0 |+ constant. (21.93)From the form of the solution we see that in two dimensions we cannot apply
the conditionF(r,r 0 )→0as|r|→∞, and in this case the constant does not
necessarily vanish.
We now return to the task of constructing the full Dirichlet Green’s function. Todo so we wish to add to the fundamental solution a solution of the homogeneous
equation (in this case Laplace’s equation) such thatG(r,r 0 )=0onS,asrequired
by (21.86) and its attendant conditions. The appropriate Green’s function is
constructed by adding to the fundamental solution ‘copies’ of itself that represent
‘image’ sources at different locationsoutsideV. Hence this approach is called the
method of images.
In summary, if we wish to solve Poisson’s equation in some regionVsubject toDirichlet boundary conditions on its surfaceSthen the procedure and argument
are as follows.
(i) To the single sourceδ(r−r 0 ) insideVadd image sourcesoutsideV∑Nn=1qnδ(r−rn) withrnoutsideV,where the positionsrnand the strengthsqnof the image sources are to be
determined as described in step (iii) below.