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(d) Use the observations in (b) and (c) above to conclude that there is a unique
point r
D
r
in the interval from r
n
to 1 such that f.r
/
D
0: Conclude that if
0
6
r
0
6
1; then 0
6
r
0
6
r
1
6
6
1:
(e) Suppose 1<r
n
: Show that
f.r
n
/>0
and
f.1/ < 0:
Use these observations to argue that f has at least one zero between 1 and r
n
:
(f) Show that f has a unique zero between 1 and r
n
; and use this to conclude that
if r
0
>1;then
r
0
>
r
1
:::
>
1:
˘
Exercise 2.7.
In a project below, we will derive further methods for solving initial
value problems of the form
u
0
.t /
D
f.
u
.t //;
u
.0/
D
u
0
;
(2.57)
where f is a given function and
u
0
is the known initial state. In this exercise, we
will derive three schemes based on formulas for numerical integration. Suppose we
want to solve (
2.57
) from t
D
0 to t
D
T .Lett
n
D
nt ,where,asusual,
t
D
T=N
and N>0is an integer.
(a) Show that
u
.t
nC1
/
D
u
.t
n
/
C
Z
t
nC1
t
n
f.
u
.t //
dt
:
(2.58)
(b) Use the trapezoidal rule to motivate the following numerical scheme
t
2
u
nC1
D
u
n
C
.f .
u
nC1
/
C
f.
u
n
// :
(2.59)
(c) Use the midpoint method to motivate the scheme
u
nC1
D
u
n
C
t f
1
2
.
u
nC1
C
u
n
/
:
(2.60)
We will derive this scheme below, using another approach. The scheme is often
called the Crank-Nicolson scheme, but it really studies a somewhat different
problem.