
17.2. Boundary Value Problems for Hyperbolic E quations wit h One Space Variable 1209
It is assumed that the functions p, p
′
x
, and q are continuous and p > 0 for x
1
≤ x ≤ x
2
.
The solution of Eq. (17.2.3.1) under the general initial conditions (17.2.1.2) and the
arbitrary linear nonhomogeneous boundary conditions (17.2.1.3)–(17.2.1.4) can be repre-
sented as the sum
w(x, t) =
Z
t
0
Z
x
2
x
1
Φ(ξ, τ )G(x, ξ, t, τ) dξ dτ
−
Z
x
2
x
1
f
0
(ξ)
∂
∂τ
G(x, ξ, t, τ )
τ=0
dξ +
Z
x
2
x
1
f
1
(ξ) + a(0)f
0
(ξ)
G(x, ξ, t, 0) dξ
+ p(x
1
)
Z
t
0
g
1
(τ)b(τ)Λ
1
(x, t, τ) dτ + p(x
2
)
Z
t
0
g
2
(τ)b(τ)Λ
2
(x, t, τ) dτ.
(17.2.3.2)
Here the Green’s function is determined by
G(x, ξ, t, τ ) =
∞
X
n=1
y
n
(x)y
n
(ξ)
ky
n
k
2
U
n
(t, τ)