Mathematical Physics Vol 1
7.7 Green formulas
377
Substituting these values into (7.269) we obtain
+ π Z
+ π Z
+ ∞ ∑ n = 1
r n R n
1 2 π
1 π
u ( r , ϕ )=
f ( t ) cos n ( t − ϕ ) d t ,
f ( t ) d t +
(7.272)
− π
− π
or
" 1 + 2
n cos n ( t − ϕ ) # f ( t ) d t .
+ π Z
+ ∞ ∑ n = 1
r R
1 2 π
u ( r , ϕ )=
(7.273)
− π
However, as
∞ ∑ n = 1
1 − c 2 c 2 − 2 c cos ϕ + 1 ,
c n cos ( n ϕ )=
1 + 2
0 ≤ c ≤ 1 ,
(7.274)
we finally obtain the solution of the Dirichlet problem for a circle
+ π Z
R 2 − r 2 R 2 − 2 rR cos ( t − ϕ )+ r 2
1 2 π
u ( r , ϕ )=
f ( t )
d t .
(7.275)
− π
This relation is known in literature also as the Poisson integral . Dirichlet problem for a sphere
In this case, the problem is reduced to finding a function u that satisfies the Laplace equation ∆ u = 0, and whose value on the sphere S is known ( u | S = f ). Observe now the sphere S , with radius R , and two points: A ( a , b , c ) in the interior of the sphere and A 1 ( a 1 , b 1 , c 1 ) out of the sphere. Let O be the center of the sphere, and point M ( x , y , z ) ∈ S (Fig. 5.4).
Figure 7.4:
Let us introduce the following notation
AM = r , A 1 M = r 1 , OA = l , OA 1 = l 1
(7.276)
A
M
A
r
r
1
l
r
α
r
A
M
O
M
A
O
l
R
1
1
1
1
Figure 7.5:
Made with FlippingBook Digital Publishing Software