
456 SECOND-ORDER PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES
leads to the two-dimensional heat equation ∂
τ
u = a
∂
ξξ
u + ∂
ηη
u
.
See also Niederer (1973) and Boyer (1974).
3.
∂w
∂t
= a
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
bx
2
+ by
2
− k
w, b > 0.
The transformation
w(x, y, t) =
1
cos
2
√
ab t
exp
√
b
2
√
a
tan
2
√
ab t
x
2
+ y
2
− kt
u(ξ, η, τ ),
ξ =
x
cos
2
√
ab t
, η =
y
cos
2
√
ab t
, τ =
√
a
2
√
b
tan
2
√
ab t
leads to the two-dimensional heat equation ∂
τ
u = ∂
ξξ
u + ∂
ηη
u.
See also Niederer (1973) and Boyer (1974).
4.
∂w
∂t
=
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
ax
−2
+ by
−2
w.
This is a special case of equation 4.3.2.7. Boyer (1976) showed that this equation admits
the separation of variables into 25 systems of coordinates for ...