Brittany's MTH 212 Blog

Project 3 Assessment

brittanymth212s10 — Wed, 05 May 2010 12:36:53 +0000

For this project we were given two first order linear differential equations

where are all constants and are unknown functions. Our goal is to use the characteristic equation to find values for the constants such that each of the following 11 cases are met:

Case 1: Two distinct real roots such that

1a) 0' title='\lambda_1, \lambda_2>0' class='latex' />

1b)

1c) 0' title='\lambda_1<0, \lambda_2>0' class='latex' />

1d) 0' title='\lambda_1=0, \lambda_2>0' class='latex' />

1e)

Case 2: Equal real roots

2a)

2b) 0' title='\lambda_1=\lambda_2>0' class='latex' />

2c)

Case 3: Complex roots

3a) 0' title='\alpha>0' class='latex' /> (spiraling out)

3b) (spiraling in)

3c) (circle)

In each case the eigenvalues are the roots for the characteristic equation .

So, using the characteristic equation and a “plug and chug” method, Laura and I were able to determine an example for each case. Then, using MATLAB we created vector field plots for each of the 11 cases. We obtained the MATLAB code from the course website.

> [x,y]=meshgrid(-1:1/10:1,-1:1/10:1);' title='>> [x,y]=meshgrid(-1:1/10:1,-1:1/10:1);' class='latex' />
> u=a*x+b*y;' title='>> u=a*x+b*y;' class='latex' />
> v=c*x+d*y;' title='>> v=c*x+d*y;' class='latex' />
> quiver(x,y,u,v,1)' title='>> quiver(x,y,u,v,1)' class='latex' />

In the code, we replaced the constants with the values found for each case. The constants we used to solve each of the cases as well as the vector field plots for each case are as follows:

1a)

0 , \lambda=2.618>0' title='\lambda=0.382>0 , \lambda=2.618>0' class='latex' />

The vector field plot of case 1a has vectors moving away from the origin and away from what looks like the line y=-x.

1b)

The vector field plot of case 1b has vectors moving along what looks like the line y=-x and look as if they are directed towards the x-axis.

1c)

0' title='\lambda=-0.414<0 , \lambda=2.414>0' class='latex' />

The vector field plot of case 1c looks similar to case 1a, where the vectors are directed away from the origin and away from what looks like the line y=-x. The vectors in this plot, however, seem start as if they were going to be directed along the line y=-x and them seem to curve away from the line as you go further into the plot.

1d)

0' title='\lambda=0 , \lambda=2>0' class='latex' />

The vector field plot of case 1d is alot like 1c and 1a. The vectors are directed away from the origin and the line y=-x.

1e)

The vector field plot of case 1e has vectors directed along the line y=x and the vectors seem to be converging towards the y-axis.

2a)

The vector field plot of case 2a is similar to 1e in that the vectors seem to be directed along the line y=x. However, this plot seems to be rotated clockwise a little bit more than the previous case.

2b)

0' title='\lambda=1>0' class='latex' />

The original vector field plot of case 2b has a rather interesting orientation. The vecotrs all are directed away from the origin and the plot is symetrical. It kind of resembles a vortex. The vectors start off aligned with the x and y-axes and then stem outward from there and point towards the corners of the graph. But seeing how we used the number zero for a few of the constants we decided to choose different constants to represent the case.

2b)

0' title='\lambda=3>0' class='latex' />

2b New

The second vector field plot of case 2b is very similar to the original, but the orientation described for the previous plot is a little less obvious in this plot. The vecotrs remain directed away from the origin and start aligned along the x and y-axes but this time the graph is not symmetrical. In the first graph there was a cross shape made by the vector tails which showed where they originated; in this plot the cross shape is still visible, but is distorted.

2c)

The vector field plot of case 2c has the vecotrs oriented in an ovular fashion, where the vectors are directed towards the y-axis as in case 1e but are pointing in the opposite directions. By the ovular fashion I mean to say that at the edges of the plot the vectors curve around a truncated version of the line y=-x.

For Cases 1 and 2 we simply plugged the constants into the characteristic equation and then put the equation into a calculator in order to solve for the eigenvalues. The method of determining the values for Case 3 were a bit different because the solutions are complex and the calculator cannot recognize them as real number values. So instead we used the quadratic equation .

We know that the will give a complex solution
Breaking the characteristic equation into its components we can see that the coefficient for is the term, the coefficient for or is the term, and is the term.
Now, we know that
In order to get a complex root so, . Simplifying this one step further, .
We then chose values such that the inequality above is true.

3a)

The vector field plot of case 3a has the vectors stemming from the origin outwards in a spiral shape.

3b)

: 3b

The vecotor field plot of case 3b has the vectors directed in a spiral directed towards the origin.

3c)

The vector field plot of case 3c makes a circular pattern slightly resembling that of case 2c in that the center of the circle is at the origin and the the circular pattern sort of revolves around the line y=-x.

When describing the majority of the vector field plots I have come to realize that many of them seem to be oriented in such a way that they involve the line y=-x and end up all looking rather similar. This is more likely than not just due to the way in which MATLAB graphs the plots because as can be seen in the calculations above all of the vector plots involve different constants and provide different eigenvalues. Although many of the plots look similar initially, upon further investigation they are slightly different in that they may have the vectors dierectied in a different direction or the plot itself may be rotated slightly.

Class March 31

brittanymth212s10 — Thu, 01 Apr 2010 02:35:15 +0000

Today we began working on project 3. So far, our methodoloy for determining the different lambda situations is pretty much plug and chug. We basically choose different values for the coefficients in the equations given, and switch it up between positive and negative coefficients and which coefficient we put a number value in for. We found equations/values for about 6 or 7 of the situations, so we are making good progress. Laura, Damola and I today used MatLab to graph the different situations, while Giuliano experimented with other programs. Next class we hope to determine the values for the majority of the remaining situations, and look at Giuliano’s programs he found because he said they were interesting…i am intrigued.

Class March 8, 10, 12

brittanymth212s10 — Fri, 12 Mar 2010 20:31:42 +0000

We started this week off doing the Euler code (which we had skipped to do the Ode 45), but by Wednesday had put that on hold to finish up our powerpoint presentation. because Giuliano and Damola had already started a presenttion for Newton’s Law of Cooling we decided that our presentation was going to be on the first project. So, on Thursday and Friday we worked on finishing the powerpoint so that by the Monday we return from spring break we will be prepared to present.

Class March 1,3,5

brittanymth212s10 — Sat, 06 Mar 2010 20:40:16 +0000

This week started off slow, but in the end we have some good results to show. On Monday, we continued our struggle with Matlab and proved to be unsuccessful in getting the Euler method code (from the class website) to provide a decent graph. We figured it was our parameters which were messing us up, but we could not find a decent combination that would make a graph where the predators and prey did not instantly die off. Then, on Wednesday, Gary introduced us to the Ode45 method. Because we were frustrated with the Euler code, we decided to move on to Ode45 and see if we could get that to work for us. (We would then move back to Euler’s code). This was not much more productive, so we asked Chris Goonan (again) for assistance. We realized that the equation we were using for the prey (from our textbook) was different from the one his group were using (which they found online). So we tried their equation and finally got a graph!!!! (I guess the equation in the book is incorrect?) On Friday, our group fooled around with the parameters a bit, making them much smaller than we were trying with the Euler code, and ended up getting what we feel is a very good graph demonstrating the relationships between predator vs. time and prey vs. time. By the end of class we were able to get a decent graph of the predator vs. prey relationship. Our graph is circular (which it should be) and looks to be a complete continuous circle (doesn’t break off like Gary said some groups did). This was very exciting because we have been extremely frustrated for the longest time and had no results to show, but in one class we were able to get two awesome ones (yay!). Next week we are going to tackle the Euler code again with our new prey equation, and hopefully we can get some graphs.

These are the equations we used in the command window for predator and prey vs. time (notice we are plotting ‘t,y(1)’ and ‘t, y(2)’)…

…and these we used for predator vs. prey (notice we are plotting ‘y’ vs ‘y’)…

This is what we had in our m-file…

As you can see all of the parameters we used are small, decimal values. At first we were using either small integer values or a combination of integers and decimal values, but once we used all small decimals (noting over 0.3) we were able to get some good graphs.

Before we left class on Friday we tried to copy the graphs we made into WordPress so we can post them in our blogs, but we were unsuccessful. So we tried putting them in a word document and taking them home and trying the process again, but they wouldn’t show the graphs on our computers (gave an error message). It’s probably because we don’t have the proper program to view the graphs on our home computers. So I tried to post them and checked to see if they showed up, but they didn’t so we because we don’t know how to post them, we have no way to show our graphs.

Class Feb. 24, 26

brittanymth212s10 — Fri, 26 Feb 2010 19:02:11 +0000

So far, we have been trying to figure out how the predator-prey equations work. On Wednesday Gary gave us a demonstration (sort of’) using Jeff’s group’s work and that sort of gave us some insight into what we need to be doing. Guiliano had talked to Gary the class before and basically knew what to do and over the past few classes showed us what he learned. We have been playing around by plugging different numbers into the equations in order to try to see exactly how the relationship between the predator and prey actually works. We did manage to overcome our previous difficulties with MATLAB and got our functions to work…we got graphs too! In another class or so, we hope to have the function of the equations figured out.

Class Feb. 22

brittanymth212s10 — Tue, 23 Feb 2010 03:11:07 +0000

Today we really did not know how to begin the project. Laura and i were diligently trying to figure out how the equations work by reading the text book section on the predator-prey models, but there were so many variables which we did not know what they were that it confused us. Guiliano talked to Gary and says he understands how to complete this project, so hopefully next class he can elaborate and we can make some progress.

Class Feb. 16, 17, 19

brittanymth212s10 — Sun, 21 Feb 2010 16:33:04 +0000

This week we were introduced to our second project. The main goal of this project is “to find numerical approximations to solutions of the non-linear systems of differential equations that arise in a variety of settings” and we were given a choice of two topics to use: a pedator-prey model or Lorenz equations. On Friday my group (Laura, Damola and Guiliano) decided to use the predator-prey models for our project. We are aware that we need to use Euler’s method to help approximate solutions to this type of differential equation because there are no exact solutions. This week was pretty much just an introduction to the project and Gary lectured for the majority of the time, so we have no work to show yet, but we are excited to get started.

First Quater Project Assessment

brittanymth212s10 — Sun, 21 Feb 2010 16:24:24 +0000

My group-mates for Project 1 were Guiliano Basile, Damola Ogunjobi and Laura Carberry.

This project asked us to investigate the behavior of differential equations using Newton’s Law of Cooling. The assignment asked us to:

Look up Newton’s law of cooling as a differential equation and to explain the parameters involved
Use the technique of separation of variables to find exact solutions to the differential equation (done by hand)
Check the solution with MATLAB’s dsolve function
Modify Newton’s law of cooling by having a variable temperature

The first part of the project required us to use a constant environmental temperature and we were told to provide examples which demonstrate how Newton’s Law works in such situations. The second part included an analysis of a dead body. We first used a temperature which varies linearly with time (room temperature cold in the morning and warm at noon) and then carried out a similar analysis in which the body is in a room for many days and the temperature varies according to a sine curve (cold at night, warmer at noon).

When we first looked at the criteria, my group was unsure of how to start the project. We felt we did not have the background knowledge of how to solve differential equations and basically did not know what to do. First we looked at the examples of Newton’s Law of Cooling provided in the textbook. The basic methodology was not hard to follow, but we got caught up when the integration factor came into play. So we decided to watch the MIT lecture on separation of variables to see if that would provide some insight on what to do; and it did. The next class we were able to complete the first few bullets of the assignment and even provided an example to model how the function works.

Newton’s Law of Cooling is as follows:

By simplifying the equation…

; let

…we were able to get an easier function to work with.

Then using separation of variables…

…we rearranged the equation to make it simpler to integrate.

This function now needed to be solved in terms of so that the function could be set equal to the change in temperature. (Remember previously: )

; let

Then, by letting we can simplify the equation:

…or…

This gave us a function of how the temperature of an object will vary in a constant environmental temperature. We also provided an example of a cup of coffee which cools according to this equation. Using (room temperature), (initial coffee temperature), and (temperature of coffee after 15 minutes) our group was able to solve the equation for the constants and . Now that all of the values used in the equation are known, we found , meaning that the temperature of the cup of coffee after 20 minutes was 182.95°F. This temperature is lower than both the initial temperature and the temperature after 15 minutes, so we know that our solution to Newton’s Law of Cooling makes sense.

The next step of the project was to check the solution in MATLAB. No one in our group knew how to use the program so we immediately knew this might be a problem for us. We decided to split up in an attempt to be more productive, so Guiliano and Damola organized our workings up to that point in a Powerpoint presentation while Laura and I toughed it out in MATLAB. With the assistance of Gary as well as Chris Goonan’s MATLAB expertise, we were able to get the separation of variables technique to work in MATLAB and solved for one of the two constants in our equation. However, when we tried to solve for the other constant we received syntax errors over and over again. Laura and I spent a considerable amount of time trying to figure out how to use the solve for that constant and how to evaluate our equation, but the program seemed to be stubborn and we could not get an answer so in the end we thought it better to move on (we are confident our solution found by hand is correct).

Next we moved on to the final step. In the second situation we needed to figure out how a temperature varying linearly with time will affect the cooling of a dead body. Laura watched a few MIT videos to get a better understanding of how to solve for this type of equation and we were able to get a function which we believe accurately models the linear temperature change. Then we attempted to solve an equation in which the temperature varies according to a sine curve (third situation), but we got stuck and did not know how to proceed. Laura and I decided that we would ask for Gary’s assistance during the next class period, but then he had an appointment, we had a snow day, and finally we had run out of time.

Overall, this project was a new, interesting and difficult experience. This was my first experience with studio-style learning, but I have to say I sort of like it. It forces us to be aware of what we need to learn and allows us to figure out how to get there (compared to being fed the information). It involves more independent work than we are used to, but it is still rewarding. Although we were unable to adequately finish all aspects of the project, I feel that my group now understands the concept fo Newton’s Law of Cooling fairly well. This law cannot be used to model how bodies decay or cool because there are too many factors involved in the process. We learned that there are a number of processes within the body which continue to function up to a certain point after death which makes the temperature’s cooling not constant and thus keeps Newton’s law from accurately. In the end, I believe we have mastered the separation of variables technique; MATLAB on the other hand…

Class Feb. 5

brittanymth212s10 — Sun, 07 Feb 2010 21:49:58 +0000

Today we split up into two groups to try and wrap things up a bit. Giuliano and Damola worked on the powerpoint presentation (which is looking good by the way) while Laura and I started the last part of the project; cooling of the dead body. We used the integration factor to get an equation for when the temperature of the room varies linearly with time (expansion of Newton’s Law of Cooling), but towrds the end of class sort of did not know how to continue. I know there is a video on MIT Courseware which might help to give some insight on where to go next…maybe i’ll watch that.

Class Feb. 3

brittanymth212s10 — Thu, 04 Feb 2010 00:46:56 +0000

Gary lectured for the first half of the class, and then we tried to make progress in MatLab but got stuck with a syntax error, so MatLab is still causing problems. I sort of wish we had taken a class on how to use the program so we would be more familiar with it, but oh well. On a brighter note, we did get a jumpstart on the powerpoint for our presentation!Friday we hope to be finished with MatLab and move on to the last few steps which deal with the cooling of dead bodies. Thats all for now…