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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
572 SECOND-ORDER HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE
Domain: −∞ < x < . Cauchy problem.
Initial conditions are prescribed:
w = f (x) at t = 0,
t
w = g(x) at t = 0.
Solution:
w(x, t)=
1
2
f(x+at) exp
bt
2a
+
1
2
f(xat) exp
bt
2a
σt
2a
exp
bx
2a
2
Z
x+at
xat
exp
2a
2
J
1
σ
p
t
2
(xξ)
2
/a
2
p
t
2
(xξ)
2
/a
2
f(ξ)
+
1
2a
exp
bx
2a
2
Z
x+at
xat
exp
2a
2
J
0
σ
p
t
2
(xξ)
2
/a
2
g(ξ)
+
1
2a
Z
t
0
Z
x+a(tτ)
xa(tτ)
exp
b(xξ)
2a
2
J
0
σ
p
(tτ)
2
(xξ)
2
/a
2
Φ(ξ, τ ) dτ,
where J
0
(z) and J
1
(z) are Bessel functions of the first kind, and σ =
1
2
|b|/a.
Domain: 0 x l. First boun dary value problem.
The following conditions are prescribed:
w = f
0
(x) at t = 0 (initial condition),
t
w =
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Publisher Resources

ISBN: 9781466581494