
7.3. Equations of the Form
∂
2
w
∂t
2
= a
2
∆
2
w − bw + Φ(x, y, t) 663
Solution:
w(x, y, t) =
∂
∂t
Z
l
1
0
Z
l
2
0
f
0
(ξ, η)G(x, y, ξ, η, t) dη dξ
+
Z
l
1
0
Z
l
2
0
f
1
(ξ, η)G(x, y, ξ, η, t) dη dξ
+ a
2
Z
t
0
Z
l
2
0
g
1
(η, τ)
∂
∂ξ
G(x, y, ξ, η, t − τ)
ξ=0
dη dτ
− a
2
Z
t
0
Z
l
2
0
g
2
(η, τ)
∂
∂ξ
G(x, y, ξ, η, t − τ)
ξ=l
1
dη dτ
− a
2
Z
t
0
Z
l
1
0
g
3
(ξ, τ )G(x, y, ξ, 0, t − τ) dξ dτ
+ a
2
Z
t
0
Z
l
1
0
g
4
(ξ, τ )G(x, y, ξ, l
2
, t − τ ) dξ dτ
+
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,
where
G(x, y, ξ, η, t) =
2
l
1
l
2
∞
X
n=1
∞
X
m=0
A
m
λ
nm
sin(p
n
x) cos(q
m
y) sin(p
n
ξ) cos(q
m
η) sin(λ
nm
t),
p
n
=
nπ
l
1
, q
m
=
mπ
l
2
, λ
nm
=
p
a
2
p
2
n
+ a
2
q
2
m
+ b, A
m
=
(
1 for m = 0,
2 for m 6= 0.
2
◦
. A rectangle is considered. The following conditions are prescribed:
w =