
436 SECOND-ORDER PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Domain: 0 ≤ x < ∞, −∞ < y < ∞. Third boundary value problem.
A half-plane is considered. The following conditions are prescribed:
w = f (x, y) at t = 0 (initial condition),
∂
x
w − kw = g(y, t) at x = 0 (boundary condition).
The solution w(x, y, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where
G(x, y, ξ, η, t) =
1
4πat
exp
−
(y − η)
2
4at
exp
−
(x − ξ)
2
4at
+ exp
−
(x + ξ)
2
4at
− 2k
Z
∞
0
exp
−
(x + ξ + s)
2
4at
− ks
ds
.
◮ Domain: 0 ≤ x < ∞, 0 ≤ y < ∞. First boundary value problem.
A quadrant of the plane is considered. The following conditions are