
406 SECOND-ORDER PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES
where
G(x, y, ξ, η, t) =
1
4πat
exp
−
(x −ξ)
2
+ (y − η)
2
4at
−exp
−
(x + ξ)
2
+ (y − η)
2
4at
.
⊙ Literature: H. S. Carslaw and J. C. Jaeger (1984).
◮ Domain: 0 ≤ x < ∞, −∞ < y < ∞. Second boundary value problem.
A half-plane is considered. The following conditions are prescribed:
w = f (x, y) at t = 0 (initial condition),
∂
x
w = g(y, t) at x = 0 (boundary condition).
Solution:
w(x, y, t) =
Z
∞
0
Z
∞
−∞
f(ξ, η) G(x, y, ξ, η, t) dη dξ
− a
Z
t
0
Z
∞
−∞
g(η, τ ) G(x, y, 0, η, t − τ) dη dτ,
where
G(x, y, ξ, η, t) =
1
4πat
exp
−
(x −ξ)
2
+ (y − η)
2
4at
+ exp
−
(x + ξ)
2
+ (y − η)
2
4at
.
◮ Domain: 0 ≤ x < ∞, −∞ < y < ∞. Third boun ...