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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
1036 HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
2
. General solution w ith Φ (x, t) = 0 and a = α
2
< 0:
w = w
1
+ w
2
,
where w
1
and w
2
are arbitrary solutions of two simpler nth-order equations
∂w
1
∂t
+ α
n
w
1
∂x
n
= 0,
∂w
2
∂t
α
n
w
2
∂x
n
= 0.
See equation 11.6.5.1 or 11.6.5.2 for information on the solution of these equations de-
pending on whether n is odd or even.
3
. Formal series solution for Φ(x, t) = 0:
w(x, t) =
X
k=0
(1)
k
a
k
t
2k
(2k)!
d
2nk
f(x)
dx
2nk
+
X
k=0
(1)
k
a
k
t
2k+ 1
(2k + 1)!
d
2nk
g(x)
dx
2nk
,
where f(x) and g(x) are arbitrary infinitely differentiable functions and d
0
f(x)/dx
0
=
f(x). This solution satisfies the initial conditions w(x, 0) = f (x) and
t
w(x, 0) = g(x).
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Publisher Resources

ISBN: 9781466581494