2.10 Semi-Infinite Rod 191
Figure 8 Graphs of the solution of the example,u(x,t)as a function ofxover
the interval 0<x< 3 b,whereb=1andT 0 =100 for convenience. The times
have been chosen so that the dimensionless timekt/b^2 takes the values 0.001, 0.01,
0.1, and 1. Whenkt/b^2 = 0 .01, the temperature nearx=b/2 has not changed
noticeably from its initial value.
Therefore, the complete solution is
u(x,t)=2
πT^0∫∞
01 −cos(λb)
λ sin(λx)exp(
−λ^2 kt)
dλ.In Fig. 8 are graphs ofu(x,t)as a function ofxforvariousvaluesoft;an
animation can be seen on the CD.
EXERCISES
- Find the solution of Eqs. (1)–(3) if the initial temperature distribution is
given by
f(x)={ 0 , 0 <x<a,
T, a<x<b,
0 , b<x.- Ve r i f y t h a tu(x,t)as given by Eq. (9) is a solution of Eqs. (1)–(3). What is
thesteady-statetemperaturedistribution? - Find the solution of Eqs. (1)–(4) iff(x)=T 0 e−αx,x>0.
- Find a formula for the solution of the problem