
996 HIGHER-ORDER PAR TIAL DIFFERENTIAL EQUATIONS
2.
∂
2
∂t
2
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+ a
2
∂
2
w
∂x
2
+ b
2
∂
2
w
∂y
2
= Φ(x, y, t).
For Φ(x, y, t) = 0, see equation 11.3.5.1.
Cauchy problem (t ≥ 0, r ∈ R
2
). Initial conditions:
w = f (r) at t = 0,
∂
t
w = g(r) at t = 0,
where r = (x, y).
Solution:
w(r, t) =
∂
∂t
Z
R
2
E
e
(r − r
′
, t)∆
′
2
f(r
′
) dV
′
+
Z
R
2
E
e
(r − r
′
, t)∆
′
2
g(r
′
) dV
′
+
Z
t
0
Z
R
2
E
e
(r − r
′
, t −τ)Φ(r
′
, τ) dV
′
dτ,
where E (r, t) is the fundamental solution given in Item 3
◦
of equation 11.3.5.1, ∆
′
2
is the
two-dimensional Laplace operator in the integration variables (x
′
, y
′
), and dV
′
= dx
′
dy
′
.
⊙ Literature: S. A. Gabov and A. G. Sveshnikov (1990).
3.
∂
2
∂t
2
∂
2
w
∂x
2
+
∂
2
w
∂y
2
−
∂
2
w
∂t
2
+
∂
2
w
∂x
2
= 0.
This ...