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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
15.2. Integral Transform Method 1173
into account the relation F{
xxxx
w} = u
4
ˇw (see Property 6 with n = 4 in Table 15.4), we arrive at
the following pro blem for a linear second-order ordina ry differential equation in t with parameter u:
ˇw
′′
tt
+ a
2
u
4
ˇw = 0,
ˇw = F (u) at t = 0, ˇw
t
= 0 at t = 0,
(15.2.2.11)
where F (u) = F {f (x)}. The solution of problem (15.2.2.11) has the form
ˇw = F (u) c os(au
2
t). (15.2.2.12)
We apply the inverse Fourier transform (15.2.2.4) to (15.2.2.12) and obtain, after easy tra nsforma-
tions,
w =
1
2π
Z
−∞
F (u) cos(au
2
t)e
iux
du
=
1
2π
Z
−∞
Z
−∞
f(ξ)e
iuξ
cos(au
2
t)e
iux
du
=
1
2π
Z
−∞
f(ξ)
Z
−∞
cos(au
2
t)e
iu(xξ)
du
=
1
π
Z
−∞
f(ξ)
Z
0
cos(
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Publisher Resources

ISBN: 9781466581494