Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 399

Therefore, there always exists a constant K for a sufficiently large R such that, by
Eq. (7.122),

≤

K
R ,

∂ u K

R 2 .

| ( ) |≤

u M

∂ n

Thus

u ∂ d S

R 2
S M 0
R

v

v ∂

u 1 K

R d S

1
R

K
R 2 d S

8 π K

R n −

≤ + =

∂ ∂ n

S M 0
R

S M 0
R

Or

u ∂
∂

1
r

d S

1

r ∂

u

lim
R → ∞

− = 0 .

r ∂ r

S M 0
R

Finally, Eq. (7.124) leads to Eq. (7.123) by letting . Equa-

tion (7.123) is called the fundamental integral formula of harmonic functions in
unbounded domains .

ε → 0and R → ∞

Green Functions in Unbounded Domains

For a harmonic function u , Eq. (7.123) reduces to

u 1

r

d S

1
4

∂
∂

1

r ∂

u

u ( M 0 )= − ( M ) − . (7.126)

π n ∂ n

∂Ω

Ω can This shows that the value of a harmonic function at any internal point of

∂Ω .Itisthe
counterpart of Eq. (7.69) for the case of an unbounded domain. Equation (7.126) is
also called the fundamental integral formula of harmonic functions in unbounded
domains .
Let v

be expressed by its values and normal derivatives on the boundary

= g ( M , M 0 ) be a harmonic function. The second Green formula thus yields

g ∂ d S

u

u ∂

g

n −

− g Δ u d Ω = 0 . (7.127)

∂ ∂ n

∂Ω Ω

Subtracting Eq. (7.127) from Eq. (7.123) leads to

u g 1

4

g ∂

d S

∂
∂

1 u

u ( M 0 )= ( M ) − +

r −

n 4 π r π ∂ n

∂Ω

g

1

+ − Δ u d Ω .

4 π r

Ω