
946 HIGHER-ORDER PAR TIAL DIFFERENTIAL EQUATIONS
An initial condition is prescribed:
w = f (x) at t = 0. (2)
One also assumes that w → 0 and ∂
x
w → 0 as |x| → ∞.
Solution:
w(x, t) =
Z
∞
−∞
E
e
(x −y, t)[f (y) − bf
′′
(y)] dy,
where the function E
e
(x, t) is defined in (1).
5
◦
. Domain: 0 ≤ x < ∞. First boundary value problem with initial condition (2).
A boundary condition is prescribed:
w = 0 at x = 0.
Solution:
w(x, t) =
Z
∞
0
E
e
(x −y, t) − E
e
(x + y, t)
[f(y) − bf
′′
(y)] dy,
where the function E
e
(x, t) is defined in (1).
6
◦
. Domain: 0 ≤ x < ∞. Second boundary value problem with initial condition (2).
A boundary condition is prescribed:
∂
x
w = 0 at x = 0.
Solution:
w(x, t)