
966 HIGHER-ORDER PAR TIAL DIFFERENTIAL EQUATIONS
◮ Solution of boundary value problems in terms of the Green’s function.
1
◦
. Consider boundary value problems on the interval 0 ≤ x ≤ l with the general initial
conditions
w = f (x) at t = 0, ∂
t
w = g(x) at t = 0
and various homogeneous boundary conditions. The solution can be represented in terms
of the Green’s function as
w(x, t) =
∂
∂t
Z
l
0
f(ξ)G(x, ξ, t) dξ +
Z
l
0
g(ξ)G(x, ξ, t) dξ
+
Z
t
0
Z
l
0
Φ(ξ, τ )G(x, ξ, t − τ) dξ dτ.
(1)
2
◦
. The Green’s functions can be evaluated from the formula
G(x, ξ, t) =
∞
X
n=1
ϕ
n
(x)ϕ
n
(ξ)
kϕ
n
k
2
sin
t
p
a
2
λ
4
n
+ k
p
a
2
λ
4
n
+ k
, (2)
where the λ
n
and ϕ
n
(x) are determined by solving the self-adjoint eigen ...