
538 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
3.
∂w
∂t
= a
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
∂
2
w
∂z
2
+
−b(x
2
+y
2
+z
2
)+ c
1
x+ c
2
y + c
3
z + s
w.
This is a special case of equation 5.3.2.3 with f
k
(t) = c
k
, g(t) = s.
4.
∂w
∂t
= a
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
∂
2
w
∂z
2
+ b
1
∂w
∂x
+ b
2
∂w
∂y
+ b
3
∂w
∂z
+ cw.
This equation governs the nonstationary temperature (concentration) field in a medium
moving with a constant velocity, provided there is volume release (absorption) of heat
proportional to temperature (concentration).
The substitution
w(x, y, z, t) = exp
A
1
x + A
2
y + A
3
z + Bt
U(x, y, z, t),
where
A
1
= −
b
1
2a
, A
2
= −
b
2
2a
, A
3
= −
b
3
2a
, B = c −
1
4a
b
2
1
+ b
2
2
+ b
2
3
,
leads to the three-dimension ...