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18.7 Steady-State Equation for a Sphere 655

with solutions that are constant multiples of
Y nm (y) = sinh(β nm y)
where

n π 2 m π 2
2
2
β nm = + .
A 2 C 2
Attempt a solution
∞ ∞

nπx mπz
u(x, y, z) = c nm sin sin sinh(β nm y).
A C
n=1 m=1
We must choose the coefficients so that
∞ ∞

nπx mπz
u(x, B, z) = f (x, z) = c nm sin sin sinh(β nm B).
A C
n=1 m=1
This is a double Fourier series for f (x, z) on 0 ≤ x ≤ A,0 ≤ z ≤ C. We have seen this type of
expansion before (Sections 17.7 and 18.5), leading us to choose
2 A C nπξ mπζ
c nm = f (ξ,ζ)sin sin dζ dξ.
AC sinh(β nm B) 0 0 A C
As usual, if nonzero data is prescribed on more than one face, split the Dirichlet problem
into a sum of problems; each of which has nonzero data on only one face.
SECTION 18.6 PROBLEMS

1. Solve 3. Solve
2
∇ u(x, y, z) = 0for 0 < x < 1,0 < y < 2π,0 < z <π,
2
∇ u(x, y, z) = 0for0 < x < 1,0 < y < 1,0 < z < 1,
u(0, y, z) = u(1, y, z) = 0,
u(0, y, z) = u(1, y, z) = 0,
u(x,0, z) = u(x, y,0) = 0,
u(x,0, z) = u(x,1, z) = 0,
2
u(x, y,π) = 1,u(x,2π, z) = xz .
u(x, y,0) = 0,u(x, y,1) = xy.
4. Solve
2. Solve
2
∇ u(x, y, z) = 0for0 < x < 1,0 < y < 2,0 < z <π,
2
∇ u(x, y, z) = 0for 0 < x < 2π,0 < y < 2π,0 < z < 1, u(x,0, z) = u(x,2, z) = 0,
u(x, y,0) = u(x, y,1) = 0, u(0, y, z) = u(x, y,π) = 0,
2
u(x,0, z) = u(x,2π, z) = 0, u(x, y,0) = x (1 − x)y(2 − y),u(1, y, z)
u(0, y, z) = 0,u(2π, y, z) = z. = sin(πy)sin(z).

18.7 Steady-State Equation for a Sphere

We will solve for the steady-state temperature distribution in a solid sphere given the temperature
at all times on the surface.
Let the sphere be centered at the origin and have a radius of R. Use spherical coordinates
(ρ,θ,ϕ) in which ρ is the distance from the origin to the point, θ is the polar angle between the
positive x-axis and the projection onto the x, y-plane of the line from the origin to the point, and
ϕ is the angle of declination from the positive z-axis to this line (Figure 18.3).

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October 14, 2010 15:27 THM/NEIL Page-655 27410_18_ch18_p641-666