
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.