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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
9.3. Helmholtz Equation
2
w + λw = Φ(x) 831
Elliptic coordinate system.
In the elliptic coordinates that are introduced by the relations
x = a cosh u cos v, y = a sinh u sin v (0 u < , 0 v < 2π, a > 0),
the Helmholtz equation is expressed as
2
w
∂u
2
+
2
w
∂v
2
+ a
2
λ(cosh
2
u cos
2
v)w = 0.
Setting w = F (u)G(v), we arrive at the following linear ordinary differential equations
for F = F (u) and G = G(v):
F
′′
+
1
2
a
2
λ cosh 2u k
F = 0, G
′′
1
2
a
2
λ cos 2v k
G = 0,
where k is the separation constant. The solutions of these equations periodic in v are given
by
F (u) =
(
Ce
n
(u, q),
Se
n
(u, q),
G(v) =
(
ce
n
(v, q),
se
n
(v, q),
q =
1
4
a
2
λ,
where Ce
n
(u, q) and Se
n
(u, q) are ...
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Publisher Resources

ISBN: 9781466581494