2.3 Example: Fixed End Temperatures 155
Figure 3 The solution of the example withT 1 =100 andT 0 =20. The function
u(x,t)is graphed as a function ofxforfourvaluesoft, chosen so that the dimen-
sionless timekt/a^2 has the values 0.001, 0.01, 0.1, and 1. Forkt/a^2 =1, the steady
state is practically achieved. See the CD.
other exponentials are still smaller. Thenw(x,t)may be well approximated
by the first term (or first few terms) of the series. Finally, ast→∞,w(x,t)
disappears completely.
EXERCISES
Also see Separation of Variables Step by Step on the CD.
- Write out the first few terms of the series forw(x,t)in Eq. (19).
- Ifk=1cm^2 /s,a=1 cm, show that aftert= 0 .5 s the other terms of the
series forware negligible compared with the first term. Sketchu(x,t)for
t=0,t= 0 .5,t= 1 .0, andt=∞.TakeT 0 =100,T 1 =300. - We can see from Eq. (19) that the dimensionless combinationsx/aand
kt/a^2 appear in the sine and exponential functions. Reformulate the partial
differential equation (8) in terms of the dimensionless variables.ξ=x/a,
τ=kt/a^2 .Setu(x,t)=U(ξ, τ ). - Sketch the functionsφ 1 ,φ 2 ,andφ 3 , and verify that they satisfy the bound-
ary conditionsφ( 0 )=0,φ(a)=0.
In Exercises 5–8, solve the problem
∂^2 w
∂x^2 =1
k∂w
∂t,^0 <x<a,^0 <t,
w( 0 ,t)= 0 ,w(a,t)= 0 , 0 <t,
w(x, 0 )=g(x), 0 <x<afor the given functiong(x).