7.2 Heat Problems 403
Usen=3andn=4. Sketch the results and explain why they vary so
much.
In Exercises 8–11, set up and solve replacement equations for the problem
stated and the given value ofn. If a computer is available, also solve forntwice
as large, and compare results.
8.d^2 u
dx^2 −^32 xu=0,^0 <x<1,
u( 0 )=0, u( 1 )=1(n=4).- d
(^2) u
dx^2
− 25 u=−25, 0 <x<1,
u( 0 )=2, u( 1 )+u′( 1 )=1(n=5).
- d
(^2) u
dx^2 +
1
1 +xdu
dx=−1,^0 <x<1,
u( 0 )=0, u( 1 )=0(n=3).- d
(^2) u
dx^2
+du
dx
−u=−x,
du
dx(^0 )=0, u(^1 )=1(n=3).
12.Use the Taylor series expansion
u(x+h)=u(x)+hu′(x)+h
2
2 u
′′(x)+h^3
6 u
( 3 )(x)+h^4
24 u
( 4 )(x)+···
withx=xiandh=± x(xi+ x=xi+ 1 ,xi− x=xi− 1 )toobtainrep-
resentations similar to Eqs. (15) and (16).
7.2 Heat Problems
In heat problems, we have two independent variablesxandt,assumedtobe
in the range 0<x<1, 0<t. A table for a functionu(x,t)should give values
at equally spaced points and times,
xi=i x, tm=m t,fori= 0 , 1 ,...,nandm= 0 , 1 ,....Here, x= 1 /n, as before. We will use a
subscript to denote position and a number in parentheses to denote the time
level for the approximation to the solution of a problem. That is,
ui(m)∼=u(xi,tm).