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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
11.6. Higher-Order Linear Equations with Constant Coefficients 1031
2.
∂w
∂t
+ a
2n+1
w
∂x
2n+1
= Φ(x, t), n = 1, 2, . . . .
1
. Particular solutions of the homogeneous equation with Φ 0:
w(x, t) = b
x
2n+1
(2n + 1)!
at
+
2n
X
k=0
c
k
x
k
,
w(x, t) = b
2x
2n+3
(2n + 3)!
ax
2
t
+ c
x
2n+2
(2n + 2)!
axt
,
w(x, t) = b exp(λx
2n+1
t) + c exp(λx +
2n+1
t),
w(x, t) = b sin
λx + (1)
n+1
2n+1
t
+ c cos
λx + (1)
n+1
2n+1
t
,
w(x, t) = exp
2n+1
t

b exp
λx
+
n
X
k=1
exp(λx cos β
k
)[b
k
cos(λx sin β
k
) + c
k
sin(λx sin β
k
)]
, β
k
=
2πk
2n + 1
,
where b, b
k
, c, c
k
, and λ are arbitrary constants.
2
. Formal series solution for Φ 0:
w(x, t) = f(x) +
X
k=1
(1)
k
a
k
t
k
k!
d
2nk+k
f(x)
dx
2nk+k
,
where f(x
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Publisher Resources

ISBN: 9781466581494