HOMEWORK #4

Due Mon. March 15 (The Ides of March)


A. Experiments with Numerical Solutions to Example 7.1 (An initial
value problem)


1. Run the script "examp7_1.m" a number of times with different time
steps.

Record the error (U-u) where U is the approximate and u the true
solution, for the times: 4 years, 10 years, and 20 years.

For each point in time (4, 10 and 20 years), make a plot of
approximation error versus time step, for the forward difference
(explicit Euler) and backward difference (implicit Euler). Describe
your findings.


2. Edit the script to solve the example using the centered difference
approximation: the centered difference with "infinity condition", and
the centered difference with "initial slope condition." (2 different
approximations).

Repeat the experiment, varying time step, and plotting
errors. Describe your findings.

Would the centered difference be appropriate if you could not invoke
an "infinity condition"?


3. Edit the script to solve the example with the Heun
("predictor-corrector") and the 4th order Runge-Kutta approximations.

Repeat the experiment, varying time step, and plotting
errors. Describe your findings.

What can you conclude about appropriate approximations for an IVP?


B. Textbook Exercises

6.4(c) Solve using the 4th order Runge-Kutta approximation.  Compare
to your analytical solution from HW#3.

For each of the following, in addition to solving it numerically,
explain why the problem is not analytically tractable.

7.4

7.5