
638 SECOND-ORDER HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES
second boundary value problem) where
G(x, y, ξ, η, t) =
4
a
∞
X
n=1
∞
X
m=1
1
E
nm
p
µ
2
n
+ ν
2
m
sin(µ
n
x + ε
n
) sin(ν
m
y + σ
m
)
× sin(µ
n
ξ + ε
n
) sin(ν
m
η + σ
m
) sin
at
p
µ
2
n
+ ν
2
m
,
ε
n
= arctan
µ
n
l
1
, σ
m
= arctan
ν
m
l
2
,
E
nm
=
l
1
+
(k
1
k
2
+ µ
2
n
)(k
1
+ k
2
)
(k
2
1
+ µ
2
n
)(k
2
2
+ µ
2
n
)
l
2
+
(k
3
k
4
+ ν
2
m
)(k
3
+ k
4
)
(k
2
3
+ ν
2
m
)(k
2
4
+ ν
2
m
)
,
where the µ
n
and ν
m
are positive roots of the transcendental equations
µ
2
− k
1
k
2
= (k
1
+ k
2
)µ cot(l
1
µ), ν
2
− k
3
k
4
= (k
3
+ k
4
)ν cot(l
2
ν).
⊙ Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
◮ Domain: 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
. Mixed boundary value problems.
1
◦
. A rectangle is considered. The following condition ...