
5.1. Heat Equation
∂w
∂t
= a∆
3
w 493
◮ Domain: 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l. First boundary value problem.
A circular cylinder of finite length is considered. The following conditions are prescribed:
w = f (r, ϕ, z) at t = 0 (initial condition),
w = g
1
(ϕ, z, t) at r = R (boundary condition),
w = g
2
(r, ϕ, t) at z = 0 (boundary condition),
w = g
3
(r, ϕ, t) at z = l (boundary condition).
Solution:
w(r, ϕ, z, t) =
Z
l
0
Z
2π
0
Z
R
0
ξf (ξ, η, ζ) G(r, ϕ, z, ξ, η, ζ, t) dξ dη dζ
− aR
Z
t
0
Z
l
0
Z
2π
0
g
1
(η, ζ, τ)
∂
∂ξ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ξ=R
dη dζ dτ
+ a
Z
t
0
Z
2π
0
Z
R
0
ξg
2
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=0
dξ dη dτ
− a
Z
t
0
Z
2π
0
Z
R
0
ξg
3
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ