
366 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
2
◦
. On passing from t, x to the new variables t, z = x −bt, we obtain a separable equation
of the form 3.8.2.2:
∂w
∂t
= a
∂
2
w
∂z
2
+
f(z) + b
∂w
∂z
.
5.
∂w
∂t
= a
∂
2
w
∂x
2
+
1
√
t
f
x
√
t
∂w
∂x
.
1
◦
. On passing from t, x to the new variables τ = ln t, ξ = x/
√
t, we obtain a separable
equation of the form 3.8.2.2:
∂w
∂τ
= a
∂
2
w
∂ξ
2
+
f(ξ) +
1
2
ξ
∂w
∂ξ
.
2
◦
. Consider the special case where the heat exchange occurs with a semiinfinite medium;
the medium has a uniform temperature w
0
at the initial instant t = 0 and the boundary x = 0
is maintained at a constant temperature w
1
all the time. In this case, the original equation
subjec ...