
548 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
5.4.2 Other Equations Containing Arbitrary Parameters
1.
∂w
∂t
= a
n
X
k=1
∂
2
w
∂x
2
k
+
c +
n
X
k=1
b
k
x
k
w.
This is a special case of equation 5.4.3.1. The transformation
w(x
1
, . . . , x
n
, t) = exp
t
n
X
k=1
b
k
x
k
+
1
3
at
3
n
X
k=1
b
2
k
+ct
u(ξ
1
, . . . , ξ
n
, t), ξ
k
= x
k
+ab
k
t
2
leads to the n-dimensional heat equation ∂
t
u = a
n
P
k=1
∂
ξ
k
ξ
k
u that is dealt with in Sec-
tion 5.4.1.
2.
∂w
∂t
= a
n
X
k=1
∂
2
w
∂x
2
k
−
c + b
n
X
k=1
x
2
k
w, b > 0.
The transformation (A is any number)
w(x
1
, . . . , x
n
, t) = u(ξ
1
, . . . , ξ
n
, τ) exp
1
2
r
b
a
n
X
k=1
x
2
k
+
n
√
ab − c
t
,
ξ
1
= x
1
exp
2
√
ab t
, . . . , ξ
n
= x
n
exp
2
√
ab t
, τ =
1
4
√
ab
exp
4
√
ab t
+ A
leads to