
518 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
Solution:
w(r, θ, ϕ, t)=
Z
2π
0
Z
π
0
Z
R
0
f(ξ, η, ζ) G(r, θ, ϕ, ξ, η, ζ, t)ξ
2
sin η dξ dη dζ
−aR
2
Z
t
0
Z
2π
0
Z
π
0
g(η, ζ, τ )
∂
∂ξ
G(r, θ, ϕ, ξ, η, ζ, t−τ )
ξ=R
sin η dη dζ dτ,
where
G(r, θ, ϕ, ξ, η, ζ, t) =
1
2πR
2
√
rξ
∞
X
n=0
∞
X
m=1
n
X
k=0
A
k
B
nmk
J
n+1/2
(λ
nm
r)J
n+1/2
(λ
nm
ξ)
× P
k
n
(cos θ)P
k
n
(cos η) cos[k(ϕ − ζ)] exp(−λ
2
nm
at),
A
k
=
(
1 for k = 0,
2 for k 6= 0,
B
nmk
=
(2n + 1)(n − k)!
(n + k)!
J
′
n+1/2
(λ
nm
R)
2
.
Here, the J
n+1/2
(r) are Bessel functions, the P
k
n
(µ) are associated Legendre functions
expressed in terms of the L egendre polynomials P
n
(µ) as follows:
P
k
n
(µ) = (1 − µ
2
)
k/2
d
k
dµ
k
P
n
(µ), P
n
(µ) =
1
n! 2
n
d
n
dµ
n
(µ
2
− 1)
n
;
and