
5.2. Heat Equation with Source
∂w
∂t
= a∆
3
w + Φ(x, y, z, t) 523
◮ Domain: −∞ < x < ∞, −∞ < y < ∞, −∞ < z < ∞. Cauchy problem.
An initial condition is prescribed:
w = f (x, y, z) at t = 0.
Solution:
w(x, y, z, t) =
Z
∞
−∞
Z
∞
−∞
Z
∞
−∞
f(ξ, η, ζ) G(x, y, z, ξ, η, ζ, t) dξ dη dζ
+
Z
t
0
Z
∞
−∞
Z
∞
−∞
Z
∞
−∞
Φ(ξ, η, ζ, τ ) G(x, y, z, ξ, η, ζ, t − τ) dξ dη dζ dτ,
where
G(x, y, z, ξ, η, ζ, t) =
1
8(πat)
3/2
exp
−
(x − ξ)
2
+ (y − η)
2
+ (z − ζ)
2
4at
.
⊙ Literature: A. G. Butkovskiy (1979).
◮ Domain: 0 ≤ x < ∞, −∞ < y < ∞, −∞ < z < ∞. First boundary value
problem.
A half-space is considered. The following conditions are prescribed:
w = f (x, y, z) at t = 0 (initial condition),
w = g(y, z, t) at