
8.4. Telegraph Equation
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∆
3
w − bw + Φ(x, y, z, t) 745
Solution:
w(x, y, z, t) =
∂
∂t
Z
l
3
0
Z
l
2
0
Z
l
1
0
f
0
(ξ, η, ζ)G(x, y, z, ξ, η, ζ, t) dξ dη dζ
+
Z
l
3
0
Z
l
2
0
Z
l
1
0
f
1
(ξ, η, ζ) + kf
0
(ξ, η, ζ)
G(x, y, z, ξ, η, ζ, t) dξ dη dζ
− a
2
Z
t
0
Z
l
3
0
Z
l
2
0
g
1
(η, ζ, τ)G(x, y, z, 0, η, ζ, t − τ ) dη dζ dτ
+ a
2
Z
t
0
Z
l
3
0
Z
l
2
0
g
2
(η, ζ, τ)G(x, y, z, l
1
, η, ζ, t − τ ) dη dζ dτ
− a
2
Z
t
0
Z
l
3
0
Z
l
1
0
g
3
(ξ, ζ, τ )G(x, y, z, ξ, 0, ζ, t −τ) dξ dζ dτ
+ a
2
Z
t
0
Z
l
3
0
Z
l
1
0
g
4
(ξ, ζ, τ )G(x, y, z, ξ, l
2
, ζ, t − τ ) dξ dζ dτ
− a
2
Z
t
0
Z
l
2
0
Z
l
1
0
g
5
(ξ, η, τ )G(x, y, z, ξ, η, 0, t − τ) dξ dη dτ
+ a
2
Z
t
0
Z
l
2
0
Z
l
1
0
g
6
(ξ, η, τ )G(x, y, z, ξ, η, l
3
, t −τ) dξ dη dτ
+
Z
t
0
Z
l
3
0
Z
l
2
0
Z
l
1
0
Φ(ξ, η, ζ, τ )G(x, y, z, ξ, η, ζ, t