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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
3.2. Heat Equation with Axial or Central Symmetry and Related Equations 301
Example 3.15. The initial temperature is unifor m, f(r) = w
0
, and the surface of the sphere is
maintained at constant temperature, g(t) = w
R
.
Solution:
w(r, t) w
R
w
0
w
R
=
2R
πr
X
n=1
(1)
n+1
n
sin
πnr
R
exp
2
n
2
t
R
2
.
The average temperature w depends on time t as follows:
w w
R
w
0
w
R
=
6
π
2
X
n=1
1
n
2
exp
2
n
2
t
R
2
, w =
1
V
Z
v
w dv,
where V is the volume of the sphere of radiu s R.
Literature: H. S. Carslaw and J. C. Jaeger (1984).
Domain: 0 r R. Second boundary value problem.
The following conditions are prescribed:
w = f (r) at t = 0 (initial condition),
r
w = g(t) at r = R (bound
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Publisher Resources

ISBN: 9781466581494