
5.1. Heat Equation
∂w
∂t
= a∆
3
w 513
Solution:
w(r, ϕ, z, t) =
Z
l
0
Z
ϕ
0
0
Z
∞
0
f(ξ, η, ζ) G(r, ϕ, z, ξ, η, ζ, t)ξ dξ dη dζ
− a
Z
t
0
Z
l
0
Z
∞
0
g
1
(ξ, ζ, τ ) G(r, ϕ, z, ξ, 0, ζ, t − τ) dξ dζ dτ
+ a
Z
t
0
Z
l
0
Z
∞
0
g
2
(ξ, ζ, τ ) G(r, ϕ, z, ξ, ϕ
0
, ζ, t − τ ) dξ dζ dτ
+ a
Z
t
0
Z
ϕ
0
0
Z
∞
0
g
3
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=0
ξ dξ dη dτ
− a
Z
t
0
Z
ϕ
0
0
Z
∞
0
g
4
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=l
ξ dξ dη dτ.
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)=
1
aϕ
0
t
exp
−
r
2
+ξ
2
4at
1
2
I
0
rξ
2at
+
∞
X
n=1
I
nπ/ϕ
0
rξ
2at
cos
nπϕ
ϕ
0
cos
nπη
ϕ
0
,
where the I
ν
(r) are modified Bessel functions.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ϕ ≤ϕ
0
, −∞ < z < ∞. First boundary ...