
4.2. Heat Equation with a Source
∂w
∂t
= a∆
2
w + Φ(x, y, t) 441
Solution:
w(x, y, t)=
Z
l
1
0
Z
l
2
0
f(ξ, η) G(x, y, ξ, η, t) dη dξ+
Z
t
0
Z
l
1
0
Z
l
2
0
Φ(ξ, η, τ) G(x, y, ξ, η, t−τ) dη dξ dτ
−a
Z
t
0
Z
l
2
0
g
1
(η, τ) G(x, y, 0, η, t−τ) dη dτ +a
Z
t
0
Z
l
2
0
g
2
(η, τ) G(x, y, l
1
, η, t−τ) dη dτ
−a
Z
t
0
Z
l
1
0
g
3
(ξ, τ) G(x, y, ξ, 0, t−τ) dξ dτ +a
Z
t
0
Z
l
1
0
g
4
(ξ, τ) G(x, y, ξ, l
2
, t−τ) dξ dτ,
where
G(x, y, ξ, η, t) =
1
l
1
l
2
1 + 2
∞
X
n=1
exp
−
π
2
n
2
at
l
2
1
cos
nπx
l
1
cos
nπξ
l
1
×
1 + 2
∞
X
m=1
exp
−
π
2
m
2
at
l
2
2
cos
mπy
l
2
cos
mπη
l
2
.
⊙ Literature: H. S. Carslaw and J. C. Jaeger (1984).
◮ Domain: 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
. Third boundary value problem.
A rectangle is considered. The following conditions are prescribed:
w = f (x,