
5.1. Heat Equation
∂w
∂t
= a∆
3
w 483
Solution:
w(x, y, z, t) =
Z
l
3
0
Z
l
2
0
Z
l
1
0
f(ξ, η, ζ) G(x, y, z, ξ, η, ζ, t) dξ dη dζ
+ a
Z
t
0
Z
l
3
0
Z
l
2
0
g
1
(η, ζ, τ)
∂
∂ξ
G(x, y, z, ξ, η, ζ, t − τ)
ξ=0
dη dζ dτ
− a
Z
t
0
Z
l
3
0
Z
l
2
0
g
2
(η, ζ, τ)
∂
∂ξ
G(x, y, z, ξ, η, ζ, t − τ)
ξ=l
1
dη dζ dτ
+ a
Z
t
0
Z
l
3
0
Z
l
1
0
g
3
(ξ, ζ, τ )
∂
∂η
G(x, y, z, ξ, η, ζ, t − τ)
η=0
dξ dζ dτ
− a
Z
t
0
Z
l
3
0
Z
l
1
0
g
4
(ξ, ζ, τ )
∂
∂η
G(x, y, z, ξ, η, ζ, t − τ)
η=l
2
dξ dζ dτ
+ a
Z
t
0
Z
l
2
0
Z
l
1
0
g
5
(ξ, η, τ )
∂
∂ζ
G(x, y, z, ξ, η, ζ, t − τ)
ζ=0
dξ dη dτ
− a
Z
t
0
Z
l
2
0
Z
l
1
0
g
6
(ξ, η, τ )
∂
∂ζ
G(x, y, z, ξ, η, ζ, t − τ)
ζ=l
3
dξ dη dτ,
where
G(x, y, z, ξ, η, ζ, t) = G
1
(x, ξ, t) G
2
(y, η, t) G
3
(z, ζ, t),
G
1
(x, ξ, t) =
2
l
1
∞
X
n=1
sin
πnx
l
1
sin
πnξ
l
1
exp
−
π
2
n
2
at
l
2
1
,
G
2
(y