2.4 Example: Insulated Bar 161
EXERCISES
- Using the initial condition
u(x, 0 )=T 1 x
a, 0 <x<a,find the solutionu(x,t)of Eqs. (1)–(3). Sketchu(x, 0 ),u(x,t)for some
t>0 (using the first three terms of the series), and the steady-state solu-
tion.- Repeat Exercise 1 using the initial condition
u(x, 0 )=T 0 +T 1(x
a) 2
, 0 <x<a.- Same as Exercise 1, but with initial condition
u(x, 0 )=
2 T 0 x
a ,^0 <x<a
2 ,
2 T 0 (a−x)
a ,a
2 <x<a.- Solve Eqs. (1)–(3) using the initial conditionu(x, 0 )=f(x),where
f(x)=
T 1 , 0 <x<a 2 ,T 2 , a
2<x<a.- Consider this heat problem, which is related to Eqs. (1)–(3):
∂^2 u
∂x^2 =
1
k∂u
∂t,^0 <x<a,^0 <t,
∂u
∂x(^0 ,t)=S^0 ,∂u
∂x(a,t)=S^1 ,^0 <t,
u(x, 0 )=f(x), 0 <x<a.
a.Show that the steady-state problem has a solution if and only ifS 0 =S 1 ,
and give a physical reason why this should be true. (Recall that the heat
fluxqis proportional to the derivative ofuwith respect tox.) Find the
steady-state solution if this condition is met.
b.Ifthesteady-statesolutionv(x) exists, show that the “transient,”
w(x,t)=u(x,t)−v(x), has the boundary conditions
∂w
∂x(^0 ,t)=^0 ,∂w
∂x(a,t)=^0 ,^0 <t.