In class, we solved
Ju
Ət
U
ди
əx
u(x, 0)
J²u
8x²¹
→ 0(x → ±∞o),
kr
→ 0(x→ ±∞o),
f(x)
=
using Fourier transforms. Here, consider the heat conduction in a half space
(0 < x <∞o) for which the boundary at x = 0) is subject to a source of heat for
t> 0
J²u
dx²
KT
Ju
Ət
u (0, t)
Uo,
u(x, t) → 0, (x→∞0),
u(t,0)
0
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)/nwhere u(r, t) is the temperature anomaly (difference of the temperature from a
constant background temperature). Show that
I
u(r, t) = Uoler fe(2/krt)]
'2√krt'
(12)
where er fc = 1- er f(x) is the complementary error function (see page 454 of
Haberman). The Laplace transform pair may be useful:
f(t) = erfc(2)
f(s) = ¹ exp(-a√/s)
8
Fig: 1
Fig: 2