
3.3. Equations Containing Power Functions and Arbitrary Parameters 321
3
◦
. The solution of the original equation in the important special case where the drop sur-
face has a time-invariant temperature w
s
and the heat exchange occurs with an infinite
medium having an initial temperature w
0
, namely,
w = w
0
at t = 0 (initial condition ),
w = w
s
at x = 0 (boundary condition ),
w → w
0
at x → ∞ (boundary condition),
is expressed in terms of the error function as follows:
w − w
s
w
0
− w
s
= erf
√
2b + 1
2
√
a
x
√
t
, erf ξ =
2
√
π
Z
ξ
0
exp
−ζ
2
dζ.
⊙ Literature: Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985).
7.
∂w
∂t
= a
∂
2
w
∂x
2
+ (bt
k
x + ct
m
)
∂w
∂x
.
This is a special case of ...