Information Technology Reference
In-Depth Information
(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.
Search WWH ::




Custom Search