
678 SECOND-ORDER HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Domain: 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
. First boundary value problem.
A rectangle is considered. The following conditions are prescribed:
w = f
0
(x, y) at t = 0 (initial condition),
∂
t
w = f
1
(x, y) at t = 0 (initial condition),
w = g
1
(y, t) at x = 0 (boundary condition),
w = g
2
(y, t) at x = l
1
(boundary condition),
w = g
3
(x, t) at y = 0 (boundary condition),
w = g
4
(x, t) at y = l
2
(boundary condition).
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) + kf
0
(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,