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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
376 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
2
. If the nonhomogeneous part of the equation can be represented in the form
Φ(x, t) =
N
X
n=1
g
n
(x)t
n
,
then there is a particular solution that is the sum of particular solutions of the form (1):
¯w(x, t) =
N
X
n=1
¯w
n
(x, t).
For example, if
Φ(x, t) = g(x)t + h(x),
where g(x) and h(x) are arbitrary functions, the original equation has a solution of the
form
¯w(x, t) = (x)
Z
x
x
0
ψ(ξ) + h(ξ)
f(ξ)
(x ξ) dξ,
ψ(x) =
Z
x
x
0
g(ξ)
f(ξ)
(x ξ) dξ, x
0
is any.
For the structure of particular solutions for other Φ(x, t), see equation 3.8.6.5, Item 3
.
By summing different solutions of the homogeneous equation (see ...
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Publisher Resources

ISBN: 9781466581494