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(g) Show that
-
If 0<x<1;then
u
goes to zero as t goes to infinity,
-
If x
D
1; then
u
D
1 for all time, and
-
If x>1,then
u
goes to infinity as t approaches
x
x
1
/:
ln.
(h) Graph the solution for x
D
0:2; 0:4; : : : ; 1:4; and discuss the stability of the
solution with respect to perturbations in the initial data.
(i) Let U.x/ be the solution at time t
D
10: Observe that
x
x
C
e
10
.1
x/
:
U.x/
D
Compute U.1/ and U.1:0000454/: Is this initial value problem stable with
respect to perturbations in the initial data?
2.4.2
More on Accuracy
In this chapter, we have been concerned with initial value problems of the form
u
0
.t /
D
f.
u
.t //;
(2.69)
u
.0/
D
u
0
;
where f is a given function and
u
0
is the given initial state. In particular, we have
studied the exponential growth model where
f.
u
/
D
au
and the logistic model where
f.
u
/
D
au
1
:
u
R
Here a and R are given parameters. We have introduced two numerical schemes:
the
explicit scheme
and the
implicit scheme
. So far we have used these terms, but
the two schemes are widely known by other names too: the explicit Euler
27
scheme,
27
Leonhard Paul Euler, 1707-1783, was a pioneering Swiss mathematician and physicist who
spent most of his life in Russia and Germany. Euler is one of the greatest scientists of all time and
made important contributions to calculus, mechanics, optics, and astronomy. He also introduced
much of the modern terminology and notation in mathematics.
