
6.1. Constant Coefficient Equations 567
Solution:
w(x, t)=
∂
∂t
Z
l
0
f
0
(ξ)G(x, ξ, t) dξ+
Z
l
0
f
1
(ξ)G(x, ξ, t) dξ+
Z
t
0
Z
l
0
Φ(ξ, τ )G(x, ξ, t−τ) dξ dτ
+a
2
Z
t
0
g
1
(τ)
h
∂
∂ξ
G(x, ξ, t−τ )
i
ξ=0
dτ+a
2
Z
t
0
g
2
(τ)G(x, l, t−τ) dτ,
where
G(x, ξ, t) =
2
al
∞
X
n=0
1
λ
n
sin(λ
n
x) sin(λ
n
ξ) sin (λ
n
at), λ
n
=
π(2n + 1)
2l
.
6.1.3 Equation of the Form
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
− bw + Φ(x, t)
This equation with Φ(x, t) ≡ 0 and b > 0 is encountered in quantum field theory and a
number of applications and is referred to as the Klein–Gordon equation.
◮ Solutions of the homogeneous equation (Φ ≡ 0).
1
◦
. Particular solutions:
w(x, t) = exp(±µt)(Ax + B), b = −µ
2
,
w(x, t) = exp(±λx)(At + B), b = a
2
λ
2
,
w(x, t) = cos(λx)[A cos ...