AIMS Exercise Set # 6
Peter J. Olver
1.
Prove that the
Midpoint Method
(10.58) is a second order method.
2.
Consider the initial value problem
du
dt
=
u
(1

u
)
,
u
(0) =
.
1
,
for the
logistic differential equation
.
(
a
) Find an explicit formula for the solution. Describe in words the behavior of the
solution for
t >
0.
(
b
) Use the Euler Method with step sizes
h
=
.
2 and
.
1 to numerically approximate the
solution on the interval [0
,
10]. Does your numerical solution behave as predicted
from part (
a
)? What is the maximal error on this interval? Can you predict the
error when
h
=
.
05? Test your prediction by running the method and computing
the error. Estimate the step size needed to compute the solution accurately to
10 decimal places (assuming no round off error)? How many steps are required?
(Just predict — no need to test it.)
(
c
) Answer part (
b
) for the Improved Euler Method.
(
d
) Answer part (
b
) for the fourth order Runge–Kutta Method.
(
e
) Discuss the behavior of the solution, both analytical and numerical, for the
alternative initial condition
u
(0) =

.
1.
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 Fall '09
 Olver
 #, Peter J. Olver

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