
468 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
or
G(x, y, z, ξ, η, ζ, t) =
1
(2
√
πat )
3
exp
−
(x − ξ)
2
+ (y − η)
2
4at
×
∞
X
n=−∞
exp
−
(z − ζ + 2nl)
2
4at
+ exp
−
(z + ζ + 2nl)
2
4at
.
⊙ Literature: H. S. Carslaw and J. C. Jaeger (1984).
◮ Domain: −∞ < x < ∞, −∞ < y < ∞, 0 ≤ z ≤ l. Third boundary value
problem.
An infinite layer is considered. The following conditions are prescribed:
w = f (x, y, z) at t = 0 (initial condition),
∂
z
w − k
1
w = g
1
(x, y, t) at z = 0 (boundary condition),
∂
z
w + k
2
w = g
2
(x, y, t) at z = l (boundary condition).
The solution w(x, y, z, t) is determined by the formula in the previous paragraph (for
the second boundary