Information Technology Reference
In-Depth Information
Extend your program from Exercise
3.1
to compute
e
F
n
e
S
n
K
n
D
and generate a plot of
K
n
K
0
K
0
for t
D
1=100 and t
D
1=500. Argue that K
n
seems to converge toward a constant, for all n,ast becomes smaller.
(e) Let t
D
10=N . Use the program you developed in (d) to compute
K
N
K
0
K
0
t
for N
D
100; 200; 300; 400; 500. Argue that
K
N
K
0
K
0
ct;
where c is independent of t .
˘
Exercise 3.3.
Consider the following system of ODEs:
u
0
D
v
3
;
u
.0/
D
u
0
;
v
0
D
(3.79)
u
3
;
v
.0/
D
v
0
:
(a) Define
r.t/
D
u
4
.t /
C
v
4
.t /;
(3.80)
and show that
r.t/
D
r.0/
(3.81)
for all t
0.
(b) Derive the following explicit finite difference scheme for system (
3.79
):
D
u
n
t
v
n
u
nC1
;
(3.82)
D
v
n
C
t
u
n
v
nC1
:
(c) Write a program that implements the scheme (
3.82
). The program should:
Accept
u
0
,
v
0
, N ,andT as input.
Compute the numerical solution defined by (
3.82
)usingt
D
T=N for
n
D
0;1;:::;N.
Plot the numerical solution of
u
and
v
both as functions of t and in the state
space.
