
528 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
4
◦
. The solutions w(x, y, z, t) of mixed boundary value problems for a rectangular paral-
lelepiped are given by the formulas in Section 5.1.1 (see the mixed boundary value prob-
lems for 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
, 0 ≤ z ≤ l
3
) with additional terms of the form (6); the
Green’s function is also taken from Section 5.1.1.
⊙ Literature: A. G. Butkovskiy (1979), H. S. C arsl aw and J. C. Jaeger (1984).
5.2.2 Problems in Cylindrical Coordinates
In the cylindrical coordinate system, the heat equation with a volume source is written as
∂w
∂t
= a
1
r
∂
∂r
r
∂w
∂r
+
1
r
2
∂
2
w
∂ϕ
2
+
∂
2
w
∂z
2
+ Φ(r, ϕ, z, t