
286 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
◮ 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 + a
Z
t
0
g(τ)Λ(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
b(ξ−x)
2a
+
c−
b
2
4a
t
exp
−
(x−ξ)
2
4at
−exp
−
(x+ξ)
2
4at
,
Λ(x, t)=
∂
∂ξ
G(x, ξ, t)
ξ=0
.
◮ Domain: 0 ≤ x < ∞. Second boundary value problem.
The following conditions are prescribed:
w = f (x) at t = 0 (initial condition),
∂
x
w = g(t) at x = 0 (boundary condition).
Substitution (1) reduces the considered equation ...