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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
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
)
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Publisher Resources

ISBN: 9781466581494