Miscellaneous Exercises 429
3.By means of the transformation mentioned in Exercise 2, a heat problem
on an annular ring is converted to
d^2 u
dx^2 +1
1 +xdu
dx=−(^1 +x),^0 <x<^1 ,
u( 0 )= 1 , u( 1 )= 0.Set up and solve replacement equations for this problem using x=
1 /4.4.The boundary value problem
1
rd
dr(
rdv
dr)
−γ^2 v= 0 , a<r<b,v(a)= 1 ,v(b)= 0 ,can be transformed into the problemd^2 u
dx^2+^1
α+xdu
dx−γ^2 L^2 u= 0 , 0 <x< 1 ,u( 0 )= 1 , u( 1 )= 0 ,whereL=b−aandα=a/L. Set up and solve replacement equations
using x= 1 /4,α=1,γL=1.5.Set up replacement equations for the heat problem in the following, and
solve fortup to 1/4, using x= 1 /4, t= 1 /32.
∂^2 u
∂x^2=∂u
∂t, 0 <x< 1 , 0 <t,u( 0 ,t)=u( 1 ,t)= 1 −e−t, 0 <t,
u(x, 0 )= 0 , 0 <x< 1.6.Same as Exercise 5, but useu( 0 ,t)=u( 1 ,t)= 1 −e−^32 (ln 2)t so that
u( 0 ,tm)= 1 −( 0. 5 )m.
7.Compare the numerical solution of the problem
∂^2 u
∂x^2=∂u
∂t, 0 <x< 1 , 0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 0 , 0 <t,
u(x, 0 )= 1 , 0 <x< 1 ,