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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
326 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
5.
∂w
∂t
= a
hh
(x + b)
2
w
∂x
2
+
∂w
∂x
ii
.
This equation describes heat transfer in a quiescent medium (solid body) in the case of
thermal diffusivity as a linear function of the space coordinate.
1
. The original equation can be rewritten in a form more suitable for applications,
∂w
∂t
= a
∂x
(x + b)
∂w
∂x
.
2
. The substitution x =
1
4
z
2
b leads to the equation
∂w
∂t
= a
2
w
∂z
2
+
1
z
∂w
∂z
,
which is considered in Section 3.2.1.
6.
∂w
∂t
= ax
2
w
∂x
2
+ bx
∂w
∂x
+ (cx + d)w.
This is a special case of equation 3.8.8.1 with f (t) = a, g(t) = b, h(t) = c, and s(t) = d.
7.
∂w
∂t
= (a
2
x + b
2
)
2
w
∂x
2
+ (a
1
x + b
1
)
∂w
∂x
+ (a
0
x + b
0
)w.
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Publisher Resources

ISBN: 9781466581494