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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
4.1. Heat Equation
∂w
∂t
= a
2
w 431
Solution:
w(r, z, t) = 2π
Z
l
0
Z
R
0
ξf (ξ, η) G(r, z, ξ, η, t)
2πaR
Z
t
0
Z
l
0
g
1
(η, τ)
∂ξ
G(r, z, ξ, η, t τ)
ξ=R
2πa
Z
t
0
Z
R
0
ξg
2
(ξ, τ ) G(r, z, ξ, 0, t τ)
+ 2πa
Z
t
0
Z
R
0
ξg
3
(ξ, τ ) G(r, z, ξ, l, t τ) dτ.
Here,
G(r, z, ξ, η, t) = G
1
(r, ξ, t) G
2
(z, η, t),
G
1
(r, ξ, t) =
1
πR
2
X
n=1
1
J
2
1
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
exp
2
n
t
R
2
,
G
2
(z, η, t) =
1
l
+
2
l
X
n=1
cos
nπz
l
cos
nπη
l
exp
an
2
π
2
t
l
2
,
where the µ
n
are positive zeros of the Bessel function, J
0
(µ
n
) = 0. The numerical values
of the first ten µ
n
are specified in Section 3.2.1 (see the first boundary value problem for
0 r R).
2
. A circular cylinder of finite length is considered. ...
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Publisher Resources

ISBN: 9781466581494