
3.1. Constant Coefficient Equations 271
where
G(x, ξ, t) =
1
2
√
πat
exp
−
(x − ξ)
2
4at
.
⊙ Literature: A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984).
◮ Domain: 0 ≤ x < ∞. First boundary value problem.
The following conditions are prescribed:
w = f (x) at t = 0 (initial condition),
w = g(t) at x = 0 (boundary condition).
Solution:
w(x, t) =
Z
∞
0
f(ξ)G(x, ξ, t) dξ +
Z
t
0
g(τ)H(x, t − τ ) dτ
+
Z
t
0
Z
∞
0
Φ(ξ, τ )G(x, ξ, t − τ) dξ dτ,
where
G(x, ξ, t) =
1
2
√
πat
exp
−
(x − ξ)
2
4at
− exp
−
(x + ξ)
2
4at
,
H(x, t) =
x
2
√
πa t
3/2
exp
−
x
2
4at
.
⊙ Literature: A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984).
◮ Domain: 0 ≤ x < ∞. Second boundary value problem.
The