2.3 Example: Fixed End Temperatures 149
- This problem describes the diffusion of a substance in a medium that is
moving with speedSto the right. The unknown functionu(x,t)is the con-
centration of the diffusing substance. Write out the steady-state problem
and solve it. (D,U,andSare constants.)
D∂(^2) u
∂x^2
=∂u
∂t
+S∂u
∂x
, 0 <x<a, 0 <t,
u( 0 ,t)=U, u(a,t)= 0 , 0 <t,
u(x, 0 )= 0 , 0 <x<a.
2.3 Example: Fixed End Temperatures
In Section 1 we saw that the temperatureu(x,t)in a uniform rod with insu-
lated material surface would be determined by the problem
∂^2 u
∂x^2 =1
k∂u
∂t,^0 <x<a,^0 <t, (1)
u( 0 ,t)=T 0 , 0 <t, (2)
u(a,t)=T 1 , 0 <t, (3)
u(x, 0 )=f(x), 0 <x<a (4)if the ends of the rod are held at fixed temperatures and if the initial tempera-
ture distribution isf(x). In Section 2 we found that the steady-state tempera-
ture distribution,
v(x)=tlim→∞u(x,t),satisfied the boundary value problem
d^2 v
dx^2 =^0 ,^0 <x<a, (5)
v( 0 )=T 0 ,v(a)=T 1. (6)Infact,wewereabletofindv(x)explicitly:
v(x)=T 0 +(T 1 −T 0 )x
a. (7)We also defined the transient temperature distribution as
w(x,t)=u(x,t)−v(x)