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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
296 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
Domain: 0 r R. Third boundary value problem.
The following conditions are prescribed:
w = f (r) at t = 0 (initial condition),
r
w + kw = g(t) at r = R (boundary condition),
|w| 6= at r = 0 (boundedness condition).
Solution:
w(r, t) =
Z
R
0
f(ξ)G(r, ξ, t) + a
Z
t
0
g(τ)G(r, R, t τ)
+
Z
t
0
Z
R
0
Φ(ξ, τ )G(r, ξ, t τ) dτ.
Here,
G(r, ξ, t) =
2
R
2
X
n=1
µ
2
n
ξ
(k
2
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
,
where the µ
n
are positive roots of the transcendental equation
µJ
1
(µ) kRJ
0
(µ) = 0.
The numerical values of the first six roots µ
n
can be found in Carslaw and Jaeger (1984).
Literature: A. V. Lyko ...
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Publisher Resources

ISBN: 9781466581494