Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 384

respectively. Therefore, the Green function plays a critical role in solving Dirichlet
problems of potential equations. Here we list analytical definitions of Green func-
tions for three kinds of boundary conditions.

Definition 1 .Let Ω be a bounded domain with a piecewise smooth boundary ∂Ω .

g ( M , M 0 ) is the solution of

⎧
⎨

Δ g = 0 , M ∈ Ω ,

(7.102)

1

⎩

g | ∂Ω =

r ,

r is distance between M 0 and M .

4 π

G ( M , M 0 ) is defined by

1

G ( M , M 0 )=

r −

g ( M , M 0 ) (7.103)

4 π

and is called the Green function of the three-dimensional Laplace operator in Ω for

the first kind of boundary-value problems .
Note that the Green function is independent of f ( M ) and F ( M ) in Eq. (7.101).

Equations (7.102) and (7.103) are equivalent to

− Δ ( , )= δ ( − ) , ∈ Ω ,

G M M 0 M M 0 M

(7.104)

( , ) | ∂Ω = .

G M M 0 0

Thus G is also called the fundamental solution of Laplace equations for the

first kind of boundary-value problems. It reflects the effect of the point source at M 0
on the solution at M

( M , M 0 )

∈ Ω .

Definition 2 .Let Ω be a bounded domain with a piecewise smooth boundary ∂Ω ,

and g ( M , M 0 ) is the solution of

⎧
⎨

Δ g = 0 , M ∈ Ω ,

∂Ω =

1
r

∂ g 1

4

∂
∂

⎩

.

∂ n π n

( , )

Then G M M 0 is defined by

1

G ( M , M 0 )=

r −

g ( M , M 0 )

4 π

or

⎧
⎨

− Δ G ( M , M 0 )= δ ( M , M 0 ) , M ∈ Ω ,

∂Ω =

(7.105)

∂ G

⎩

0

∂ n

and is called the Green function of the three-dimensional Laplace operator in Ω for

the second kind of boundary-value problems .