
8.6. Equations with n Space Variables 775
Here,
G(x, y, t) =
1
l
1
l
2
. . . l
n
∞
X
s
1
=0
∞
X
s
2
=0
. . .
∞
X
s
n
=0
A
s
1
A
s
2
. . . A
s
n
cos(λ
s
1
x
1
) cos (λ
s
2
x
2
) . . . cos(λ
s
n
x
n
)
× cos(λ
s
1
y
1
) cos(λ
s
2
y
2
) . . . cos(λ
s
n
y
n
)
sin
t
p
a
2
(λ
2
s
1
+ ··· + λ
2
s
n
) + b
p
a
2
(λ
2
s
1
+ ··· + λ
2
s
n
) + b
,
where
λ
s
1
=
s
1
π
l
1
, λ
s
2
=
s
2
π
l
2
, . . . , λ
s
n
=
s
n
π
l
n
; A
s
m
=
(
1 fo r s
m
= 0,
2 fo r s
m
6= 0,
m = 1, 2, . . . , n.
◮ Domain: V = {0 ≤ x
k
≤ l
k
; k = 1, . . . , n}. Third boundary value problem.
The following conditions are prescribed:
w = f
0
(x) at t = 0 (initial condition),
∂
t
w = f
1
(x) at t = 0 (initial condition),
∂
x
k
w − b
k
w = g
k
(x, t) at x
k
= 0 (boundary conditions),
∂
x
k
w + c
k
w = h
k
(x, t) at x
k
= l
k
(boundary conditions).
The