
5.1. Heat Equation
∂w
∂t
= a∆
3
w 503
Here,
G(r, ϕ, z, ξ, η, ζ, t) = G
1
(r, ϕ, ξ, η, t)
2
l
∞
X
n=1
sin
nπz
l
sin
nπζ
l
exp
−
an
2
π
2
t
l
2
,
G
1
(r, ϕ, ξ, η, t) =
π
2
∞
X
n=0
∞
X
m=1
A
n
B
nm
Z
n
(µ
nm
r)Z
n
(µ
nm
ξ) cos[n(ϕ − η)] exp(−µ
2
nm
at),
A
n
=
(
1/2 for n = 0,
1 for n 6= 0,
B
nm
=
µ
2
nm
J
2
n
(µ
nm
R
2
)
J
2
n
(µ
nm
R
1
) −J
2
n
(µ
nm
R
2
)
,
Z
n
(µ
nm
r) = J
n
(µ
nm
R
1
)Y
n
(µ
nm
r) − Y
n
(µ
nm
R
1
)J
n
(µ
nm
r),
where the J
n
(r) and Y
n
(r) are Bessel functions and the µ
nm
are positive roots of the
transcendental equation
J
n
(µR
1
)Y
n
(µR
2
) − Y
n
(µR
1
)J
n
(µR
2
) = 0.
◮ Domain: R
1
≤ r ≤R
2
, 0 ≤ ϕ ≤2π, 0 ≤ z ≤ l. Second boundary value problem.
A circular cylinder of finite length is considered. The following conditions are prescribed:
w = f (r, ϕ, z) at