
4.1. Heat Equation
∂w
∂t
= a∆
2
w 411
where
G(x, y, ξ, η, t) = G
1
(x, ξ, t) G
2
(y, η, t),
G
1
(x, ξ, t) =
1
l
+
2
l
∞
X
n=1
cos
nπx
l
cos
nπξ
l
exp
−
an
2
π
2
t
l
2
,
G
2
(y, η, t) =
1
2
√
πat
exp
−
(y − η)
2
4at
+ exp
−
(y + η)
2
4at
.
◮ Domain: 0 ≤ x ≤ l, 0 ≤ y < ∞. Third boundary value problem.
A semiinfinite strip is considered. The following conditions are prescribed:
w = f (x, y) at t = 0 (initial condition),
∂
x
w − k
1
w = g
1
(y, t) at x = 0 (boundary condition),
∂
x
w + k
2
w = g
2
(y, t) at x = l (boundary condition),
∂
y
w − k
3
w = g
3
(x, t) at y = 0 (boundary condition).
The solution w(x, y, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where ...