
1064 SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
12.3.2 Systems of Hyperbolic or Elliptic Equations
1.
∂
2
u
∂t
2
= k
∂
2
u
∂x
2
+ a
1
u + b
1
w,
∂
2
w
∂t
2
= k
∂
2
w
∂x
2
+ a
2
u + b
2
w.
Constant coefficient second-order linear system of hyperbolic type.
Solution:
u =
a
1
− λ
2
a
2
(λ
1
−λ
2
)
θ
1
−
a
1
− λ
1
a
2
(λ
1
− λ
2
)
θ
2
, w =
1
λ
1
− λ
2
θ
1
− θ
2
,
where λ
1
and λ
2
are roots of the quadratic equation
λ
2
− (a
1
+ b
2
)λ + a
1
b
2
− a
2
b
1
= 0
and the functions θ
n
= θ
n
(x, t) satisfy the independent linear Klein–Gordon equations
∂
2
θ
1
∂t
2
= k
∂
2
θ
1
∂x
2
+ λ
1
θ
1
,
∂
2
θ
2
∂t
2
= k
∂
2
θ
2
∂x
2
+ λ
2
θ
2
.
2.
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= a
1
u + b
1
w,
∂
2
w
∂x
2
+
∂
2
w
∂y
2
= a
2
u + b
2
w.
Constant coefficient second-order linear system of elliptic type.
Solution:
u =
a
1
− λ
2
a
2
(λ
1
−λ
2
)