Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 361

where x = cos θ , − 1 ≤ x ≤ 1. Note that

x 2 d n − 1

1 n d x 1 n

1

2 1
−

1

d x n − 1 x 2

x d n − 1

d x n − 1 x 2

1
2 n n !

x 2 P n ( x ) d x = −

1 −

−

− 1 1

−

x

1 n d x 1 n

1 1

d x n − 2 x 2 d x n − 2 x 2

d n − 2 d n − 2

2
2 n n !

= − −

1 −

−

− 1

−

1 n

d x n − 3 x 2

d n − 3 2

2 n n !

1
− 1 =

= − 0 , when n ≥ 3 .

Thus we obtain

⎧
⎨

1
3 ,

n = 0 ,

1

2 n + 1

x 2 P n (

2
3

C n = x ) d x = , n = 2 ,

⎩

2

− 1

0 , n = 0 , 2 .

Finally, we have

r 2 cos 2

1
3 +

2
3 r 2 P 2 (

1
3 +

1
3

u ( r , θ )= x )= θ − .

Example 4. Consider steady heat conduction in a sphere of radius R . The tempera-
ture is kept at a constant v 0 on the upper half of the spherical boundary surface and
at zero on the lower half. Find the steady temperature distribution in the sphere.

Solution . By the given conditions, the temperature T cannot depend on ϕ in a spher-

ical coordinate system ( r , θ , ϕ ) . It must satisfy

r 2 ∂ sin

⎧
⎨

r 2 ∂

1 T

∂

1
r 2 sin

∂
∂θ

θ ∂

T
∂θ

+ = 0 ,

∂ r r θ

< < , < θ < π ,

0 r R 0

(7.61)

⎧
⎨

⎩

≤ θ < 2 ,

,

v 0 0

T ( R , θ )=

2

0 , < θ ≤ π .

⎩

By following a similar approach as that in Example 3, we can obtain the solution of
the Laplace equation in PDS (7.61)

∞
n = 0 C n r n P n ( cos θ ) ,
where the C n are constants. Applying the boundary condition at r

T ( r , θ )=

=

R yields

v 0

∞
n = 0 C n R n P n ( x )=

, 0 < x ≤ 1 ,

,− ≤ < ,

0 1 x 0