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In-Depth Information
subject to
u
D
u
.x; t I k/
satisfying
u
t
D .k
u
x
/
x
for
x 2 .0; 1/; t > 0;
k.0/
u
x
.0; t / D 0
for
t>0;
k.1/
u
x
.1; t / D 0
for
t>0;
u
.x; 0/ D f.x/
for
x 2 .0; 1/:
Here,
h
1
.t /
and
h
2
.t /
represent the measurements made at the endpoints
x D 0
and
x D 1
, respectively. We assume that the initial condition
f.x/
is given and that
@
u
=@x D 0
at the boundaries.
This is certainly a very difficult problem. To solve it, a number of mathemat-
ical and computational techniques developed throughout the last decades must be
employed. This exceeds the ambitions of the present text, but we encourage the
reader to carefully evaluate his or her understanding of the output least squares
method by formulating such an approach for a problem involving, e.g., a system of
ordinary differential equations.
9.4
Exercises
Exercise 9.1.
(a) Use the bisection method to solve (9.20).
(b) Create a plot similar to that shown in Fig. 5.13. That is, visualize the actual total
world population and the graph of the function
2:555 10
9
e
at
,where
a
is the
growth rate computed in (a), in the same figure.
(c) Compare the actual population in the year
2000
with that suggested/predicted
by the model
2:555 10
9
e
at
. How large is the error in the prediction?
(d) Solve (9.20) with Newton's method.
˘
Exercise 9.2.
(a) Apply Newton's method to estimate the initial condition
r
0
and
the growth rate
a
by solving the nonlinear system of algebraic equations (
9.15
)and
(
9.16
). You can use the population in the year
1950
and the growth rate estimated in
Exercise
9.1
as the initial guess for the Newton iteration.
(b) Redo assignments (b) and (c) in Exercise
9.1
with the parameters estimated by
solving the system (
9.15
)and(
9.16
).
˘
Exercise 9.3.
The model for logistic growth (
9.3
)and(
9.4
) involves three parame-
ters: the size of the initial population
r
0
, the growth rate
a
, and the carrying capacity
R
.WewanttouseTable5.4, which contains the number of people on Earth for the
period 1990-2000, to estimate these quantities. To this end, let
t D 0
correspond to
1990
,
t D 1
correspond to
1991
, and so on, and let
d
0
;d
1
; :::; d
10
represent the
total world population in
1990; 1991; : : : ; 2000
, respectively.
