Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 397

7.7.3 Boundary-Value Problems in Unbounded Domains

We have proven the third Green formula (Eq. (7.65)) and the formula (7.109) for
a bounded domain , and used them in some examples with unbounded domains

in Section 7.7.2. Here we discuss the conditions for their validation for unbounded
domains.

Ω

Fundamental Integral Formula

Ω be an unbounded domain with a piecewise smooth boundary
∂Ω . Suppose that u

Theorem 1. Let

¯

C 1 Ω ) ∩ C 2

( Ω )

( )= ( , , ) ∈ (

,andas r OM → ∞ M u x y z

O 1
r OM

O 1
r OM

∂ u

u ( M )= ,

n =

. (7.122)

∂

Here the r OM is the distance between the origin O and point M ,and
¯

Ω = Ω ∪ ∂Ω .

Thus the third Green formula is valid,

u 1

r

d S

1
4

) ∂
∂

1

r ∂

u 1

4

Δ u

r

u ( M 0 )= − ( M − − d Ω , (7.123)

π n ∂ n π

∂Ω Ω

where M 0 ∈ Ω and r is the distance between M 0 and M .
Proof. Consider two spherical surfaces S M 0 and S M 0

R

with sufficiently small ε and

ε

∂Ω is contained in the region of boundary S M 0 sufficiently large R such that the and

ε

S M R (Fig. 7.4). Let Ω ∗ be the region formed by S M 0 ∂Ω and S M R .The Ω ∗ reduces

,

ε

1
r M 0 M =

1
r , applying the second Green formula

Ω as

to ε → 0and R → ∞ . With v =

yields

u ∂
∂

1
r

d S

1

r ∂

u Δ u

r

− = − d Ω . (7.124)

n ∂ n

Ω ∗

S M 0 S M 0

R

∂Ω ∪

ε ∪

Note that

u ∂
∂

1
r

d S 1

r

d S

1

r ∂

u

u ∂
∂

1
ε

∂ u

− = − −

n d S n ∂ n r ∂

S M 0
ε

S M 0
ε

S M 0
ε

1
ε

− ∂

u

M ξ ) − ∂

u

= u d S

n 4

πε = 4 π u (

n 4

πε ,

2

∂ ∂

S M 0
ε