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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
4.2. Heat Equation with a Source
∂w
∂t
= a
2
w + Φ(x, y, t) 451
The solution w(r, z, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where
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
µ
2
n
(k
2
1
R
2
+µ
2
n
)J
2
0
(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) =
X
m=1
ϕ
m
(z)ϕ
m
(η)
kϕ
m
k
2
exp(
2
m
t), ϕ
m
(z) = cos(λ
m
z)+
k
2
λ
m
sin(λ
m
z),
kϕ
m
k
2
=
k
3
2λ
2
m
λ
2
m
+k
2
2
λ
2
m
+k
2
3
+
k
2
2λ
2
m
+
l
2
1+
k
2
2
λ
2
m
,
and the µ
n
and λ
m
are positive roots of the transcendental equations
µJ
1
(µ) k
1
RJ
0
(µ) = 0,
tan(λl)
λ
=
k
2
+ k
3
λ
2
k
2
k
3
.
Domain: 0 r R, 0 z l. Mixed boundary value problems.
1
. A circular cylinder of finite length is considere
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Publisher Resources

ISBN: 9781466581494