
5.1. Heat Equation
∂w
∂t
= a∆
3
w 473
Solution:
w(x, y, z, t) =
Z
∞
0
Z
∞
0
Z
∞
0
G(x, y, z, ξ, η, ζ, t)f(ξ, η, ζ) dξ dη dζ
−a
Z
t
0
Z
∞
0
Z
∞
0
g
1
(η, ζ, τ) G(x, y, z, 0, η, ζ, t − τ) dη dζ dτ
−a
Z
t
0
Z
∞
0
Z
∞
0
g
2
(ξ, ζ, τ ) G(x, y, z, ξ, 0, ζ, t − τ) dξ dζ dτ
−a
Z
t
0
Z
∞
0
Z
∞
0
g
3
(ξ, η, τ ) G(x, y, z, ξ, η, 0, t − τ) dξ dη dτ,
where
G(x, y, z, ξ, η, ζ, t) =
1
2
√
πat
3
H(x, ξ, t)H(y, η, t)H(z, ζ, t),
H(x, ξ, t) = exp
−
(x − ξ)
2
4at
+ exp
−
(x + ξ)
2
4at
.
◮ Domain: 0 ≤ x < ∞, 0 ≤ y < ∞, 0 ≤ z < ∞ . Third boundary value problem.
An octant is considered. The following conditions are prescribed:
w = f (x, y, z) at t = 0 (initial condition),
∂
x
w − k
1
w = g
1
(y, z, t) at x = 0 (boundary condition),
∂
y
w − k
2