2. Laplace transform solution of the diffusion equation

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