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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
84 Applied calculus of variations for engineers
Finally the remaining equation is of the Bessel kind:
r
2
d
2
u
1
dr
2
+ r
du
1
dr
+(k
2
r
2
m
2
)u
1
=0.
The solution of such a differential equation when m is not an integer is of the
form
u
1
(r)=c
5
J
m
(kr)+c
6
J
m
(kr),
where J are the Bessel functions of the first kind, defined by the formula
J
m
(x)=
n=0
(1)
n
n!(n + m)!
(
x
2
)
m+2n
,
This is a convergent series for any x = kr value. The J
m
function in the
expression is simply defined by
J
m
(x)=(1)
m
J
m
(x).
However, in our case m is an integer and then the solution is
u
1
(r)=c
5
J
m
(kr)+c
6
Y
m
(kr).
The Bessel function of the second kind is defined as
Y
m
(x) = lim
pm
cos()J
p
(x) J
p
(x)
sin()
.
The ...
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Publisher Resources

ISBN: 9781482253597