Use the method of separation of variables to show that the 1-D heat conduction problem: find u = u(x,t) so that
u
t
(x,t) - a 2 u
x2
(x,t)
=
0,  0 < x < l,  t 0
u(0,t)
=
0,  t 0
u(l,t)
=
0,  t 0
u(x,0)
=
f(x),  0 x l
has the solution
u(x,t) =

n = 1 
cn sin n px
l
exp

-a

n p
l


2

 
t

where
cn = 2
l

l

0 
f(x) sin n px
l
dx.


File translated from TEX by TTH, version 2.22.
On 13 Jun 1999, 00:56.