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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
582 SECOND-ORDER HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE
Reduction to a constant coefficient equation.
The substitution u(r, t) = rw(r, t) leads to the constant coefficient equation
2
u
∂t
2
= a
2
2
u
∂r
2
,
which is discussed in Section 6.1.1.
Domain: 0 r < . Cauchy problem.
Initial conditions are prescribed:
w = f (r) at t = 0,
t
w = g(r) at t = 0.
Solution:
w(r, t) =
1
2r
(r at)f
|r at|
+ (r + at)f
|r + at|

+
1
2ar
Z
r+at
rat
ξg
|ξ|
dξ.
Solution at the center r = 0:
w(0, t) = atf
(at) + f(at) + tg(at).
Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
Domain: 0 r R. First boundary value problem.
The following conditions are
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Publisher Resources

ISBN: 9781466581494