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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
10.3. Helmholtz Equation
3
w + λw = Φ(x) 929
10.3.6 O ther Orthogonal Curvilinear Coordinates
The homogenous three-dimensional Helmholtz equation admits separation of variables in
the eleven orthogonal systems of coordinates listed in Table 10.4.
For the parabolic cylindrical system of coordinates, the multipliers f and g are ex-
pressed in terms of the parabolic cylinder functions as
f(ξ)=A
1
D
µ1/2
(σξ)+A
2
D
µ1/2
(σξ), g(η)=B
1
D
µ1/2
(ση)+B
2
D
µ1/2
(ση),
µ=
1
2
β(k
2
λ)
1/2
, σ=
4(k
2
λ)
1/4
,
where A
1
, B
1
, A
2
, and B
2
are arbitrary constants.
For the elliptic cylindrical system of coordinates, the functions f and g are determined
by the modified Mathieu equation ...
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Publisher Resources

ISBN: 9781466581494