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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
826 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
Domain: 0 r R. Third boundary value problem.
A circle is considered. A boundary condition is prescribed:
r
w + kw = 0 at r = R.
Eigenvalues:
λ
nm
=
µ
2
nm
R
2
; n = 0, 1, 2, . . . ; m = 1, 2, 3, . . .
Here, the µ
nm
is the mth root of the transcendental equation µJ
n
(µ) + kRJ
n
(µ) = 0.
Eigenfunctions:
w
(1)
nm
= J
n
r
p
λ
nm
cos nϕ, w
(2)
nm
= J
n
r
p
λ
nm
sin nϕ.
The square of the norm of an eigenfunction is given by
kw
(1)
nm
k
2
=kw
(2)
nm
k
2
=
πR
2
(1+δ
n0
)
2µ
2
nm
(k
2
R
2
+µ
2
nm
n
2
)[J
n
(µ
nm
)]
2
, δ
ij
=
(
1 for i=j,
0 for i6=j.
Literature: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin et al. (1964), B. M. Budak, A. A. Samarskii,
and ...
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Publisher Resources

ISBN: 9781466581494