Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 398

Fig. 7.4 Spherical surfaces S M 0
ε

and S M 0
R

where u and ∂

u

n are the mean-values of u and ∂

u

on S M 0

, respectively. Since

ε

∂ ∂ n

u ( M ξ ) → u ( M 0 ) as ε → 0, we obtain

u ∂
∂

1
r

d S

1

r ∂

u

− → 4 π u ( M 0 ) , as ε → 0 .

n ∂ n

S M 0
ε

Also,

u ∂ d S u ∂

∂

1
r

d S

v

v ∂

u 1

r ∂

u

n −

= −

∂ ∂ n r ∂ r

S M 0
R

S M 0
R

R 2
S M 0
R

1 1

R

∂ u

= − u d S −

r d S

.

∂

S M 0
R

Consider now OMM 0 where the O is the origin of the coordinate system, M 0 is a

fixed point inside

Δ

Ω ,and M is a point on S M 0
R

.Wehave

r OM 0

r M 0 M

r OM + r OM 0 > r M 0 M or 1 +

r OM > r OM .

(7.125)

lim r M 0 M
r OM

Therefore, as r OM → ∞ ,1 ≥ , r M 0 M = R . Since, on the other hand, r M 0 M >

r OM 0 , we have lim r M 0 M r OM −

r OM ≥

1. Therefore we obtain

R
r OM =

lim
r OM → ∞

1 .