
11.4. Three- and n-Dimensional Nonstationary Fourth-Order Equations 1001
where
ϕ
nmk
(t) =
−
2
s
nmk
λ
nmk
Z
t
0
A
nmk
(τ) exp
−
1
2σ
(t −τ)
sin
s
nmk
2σ
(t −τ)
dτ
if 4νσλ
nmk
− 1 = s
2
nmk
> 0,
−
2
s
nmk
λ
nmk
Z
t
0
A
nmk
(τ) exp
−
1
2σ
(t −τ)
sinh
s
nmk
2σ
(t − τ)
dτ
if 4νσλ
nmk
− 1 = −s
2
nmk
< 0,
−
1
σλ
nmk
Z
t
0
A
nmk
(τ) exp
−
1
2σ
(t − τ)
(t − τ) dτ
if 4νσλ
nmk
− 1 = 0,
u
nmk
(x) = sin
πnx
a
sin
πmy
b
sin
πkz
c
,
λ
nmk
= π
2
n
2
a
2
+
m
2
b
2
+
k
2
c
2
, A
nmk
(t) =
8
abc
Z
V
f(x, t)u
nmk
(x) dV.
4.
∂
2
∂t
2
− c
2
1
∆
∂
2
∂t
2
− c
2
2
∆
w = Φ(x, t).
Here x = (x, y, z) and ∆ is the three-dimensional Laplace operator. S imilar equations
occur in elasticity theory (see Section 12.6.3).
1
◦
. The general solution of the ...