Math 2250 - 002
Maple
Project #2
Due: 9/20/11
Math 2250-002
Second Software (Maple) Assignment
Numerical Approximation for Solutions to Differential Equations
If you choose to do this project using Maple, begin by downloading this document and opening it
from Maple. A shortcut is to use the "open URL" option in the "File" menu item.
Start File:
http://www.math.utah.edu/~korevaar/2250fall11/maplehw2.mw
Also recall that there is an explanation of several important Maple commands in the file posted at
Commands: http://www.math.utah.edu/~camacho/teaching/fall2011/Maple_Commands.pdf
In this project you will be using some small "for loops" with "conditionals" to do the iterations. I
would suggest you investigate the examples in the help documentation. Recall that to see the
help documentation for a specific command you can use the following syntax:
>
>
Wikipedia has good explanations of these basic programming ideas - just google "for loop
wikipedia" or "conditional (programming) wikipedia". You will be using the Euler and RungeKutta numerical techniques. Sections 2.4 and 2.6 contain pseudocode for these algorithms.
There are Matlab and TI-85 implementations at the ends of these sections. There is a Maple file
illustrating the how to implement Euler's method in the Maple posted online at
Euler's Method: http://www.math.utah.edu/~camacho/teaching/fall2011/Euler.pdf
You will be using the Euler and Runge-Kutta numerical algorithms. Sections 2.4 and 2.6 contain
pseudocode1 for these algorithms. There are Matlab and TI-85 implementations at the ends of
these sections.
You will create a Maple document containing a mixture of text and mathematics, in which you
answer various mathematical questions.
1
Psuedocode is an organized way to describe a computer program or algorithm using words rather than actual
code. You are essentially describing what your code is doing in the order that the algorithm is carried out.
Math 2250 - 002
Maple
Project #2
Due: 9/20/11
Important directions for submission for Math 2250-2: You will hand in a hard copy of all of
your work for this assignment. There are three things you need to include in your project report.
1. Using the start file referenced above, insert your Maple commands into the appropriate lines
within the file. Make sure that you suppress any output that is not explicitly asked for in each
question, and display the output that you are asked to display. Print this Maple document and
write your name on the front page.
2. Answer each of the questions with any relevant explanations, equations, functions, graphs, or
tables. This part of the report may either be handwritten or typed. You do not need to show all
of your work for how you arrived at each answer, simply present your answers. Your work was
done by your Maple code.
Staple all of these items together, in the order listed above, and hand in your work at the
beginning of class on Tuesday, September 20th. You may turn in the project early if you would
like.
Math 2250 - 002
Maple
Project #2
Due: 9/20/11
Name:______________________
Consider the IVP
.
You solved this IVP in the first Maple project - the solution is
This problem and its solution could be modeling a linear drag problem for a parachutist who
drops out of a helicopter at time
and immediately pulls his ripcord. In this model we
measure positive distance in the downwards direction; the acceleration of gravity is
, and
the drag of the parachute adds the deceleration term
out what percentage of the terminal velocity of
. In the first project you figured
is obtained after
and
seconds.
1.
Use Euler's method to approximate the solution to the IVP on the interval
follows:
, as
(a) Use step size
. Write your "for" loop so that Maple only outputs the approximate
values at
. What is the relative error of your numerical estimates as compared to the exact
solution values, at
? (Recall that relative error is the error divided by the exact value,
expressed as a percentage)
(b) Now use step size
, display at the same
about the relative error as in part (a)
values and answer the same questions
(c) Now use step size
and display at the same
about the relative error as in parts (a) and (b).
2.
values and answer the same questions
Use the Runga-Kutta method to approximate the solution to the IVP on the interval
. Have Matlab output only the approximate values at
. Use step size
and
calculate the relative error of your numerical estimates. How does the accuracy of Runge-Kutta
with step size
compare to the accuracy of Euler's Method with the three step sizes you
used in question 1?
Math 2250 - 002
Maple
Project #2
Due: 9/20/11
3.
The wind resistance provided by a parachute is not really a linear deceleration. Suppose
that for a certain type of parachute and for a person of the same mass as in the original model,
the acceleration equation is more effectively modeled by
(a) What is the terminal velocity in this model? [You will have difficulty solving this
algebraically, it will be helpful to use the solve() command in Maple. For instructions in how to
use Maple's solve command see the Maple Commands2 file]
(b) This is differential equation does NOT have an elementary solution. Use Runge-Kutta with
step sizes
and
to estimate the solution values at
and
seconds. Based on a
comparison of the estimated values at these two values, how confident are you that either one
of them provides a good estimate for the actual values
and
? Explain your reasoning.
(c) Based on your work in part (b), what percent of the terminal velocity is attained at
seconds and at
seconds?
2
http://www.math.utah.edu/~camacho/teaching/fall2011/Maple_Commands.pdf