
5.4. Equations with n Space Variables 553
Solution:
w(x, t) =
Z
t
0
Z
V
Φ(y, τ) G(x, y, t, τ) dV
y
dτ +
Z
V
f(y) G(x, y, t, 0) dV
y
+
n
X
k=1
Z
t
0
Z
S
(k)
a
k
(α
k
, τ)
g
k
(y, τ )
∂
∂y
k
G(x, y, t, τ )
y
k
=α
k
dS
(k)
y
dτ
−
n
X
k=1
Z
t
0
Z
S
(k)
a
k
(β
k
, τ)
h
k
(y, τ)
∂
∂y
k
G(x, y, t, τ )
y
k
=β
k
dS
(k)
y
dτ,
where
dV
y
= dy
1
dy
2
. . . dy
n
, dS
(k)
y
= dy
1
. . . dy
k−1
dy
k+1
. . . dy
n
,
S
(k)
= {α
m
≤ y
m
≤ β
m
for m = 1, . . . , k−1, k+1, . . . , n}.
The Green’s function can be represented in the product form
G(x, y, t, τ ) =
n
Y
k=1
G
k
(x
k
, y
k
, t, τ). (1)
Here, the G
k
= G
k
(x
k
, y
k
, t, τ) are auxiliary Green’s functions that, for t > τ ≥ 0, satisfy
the one-dimensional linear homogeneous equations
∂G
k
∂t
− a
k
(x
k
, t)
∂
2
G
k
∂x
2
k
− b
k
(x
k
, t)
∂G
k
∂