onestep
There have been numerous problems downloading programs for TI graphing calculators using the method above. If that method works, it is very clean. Sometimes, however, it doesn't work. In that case we can fall back on an older method, uuencoding, that is less user-friendly but more reliable. Click here for more information about this method. Then click here for a uuencoded (text file).
ndsolve
There have been numerous problems downloading programs for TI graphing calculators using the method above. If that method works, it is very clean. Sometimes, however, it doesn't work. In that case we can fall back on an older method, uuencoding, that is less user-friendly but more reliable. Click here for more information about this method. Then click here for a uuencoded (text file).
foobar
There have been numerous problems downloading programs for TI graphing calculators using the method above. If that method works, it is very clean. Sometimes, however, it doesn't work. In that case we can fall back on an older method, uuencoding, that is less user-friendly but more reliable. Click here for more information about this method. Then click here for a uuencoded (text file).
These programs will enable you to find numerical estimates for solutions to systems of differential equations like the ones in this module. We use them to approximate the solution of the following system.
B1' = 0.25 * (E - 2 * B1 + B2)
B2' =
0.25 * (B1 - 2 * B2 + B3)
B3' =
0.25 * (B2 - 2 * B3 + B4)
I' = 0.25 * (B3 - I)
B1(0) = 1
B2(0) = 2
B3(0) = 3
I(0) = 4
In this system E(t) is the exterior temperature and will be given by the function
E(t) = sin (pi t / 12)
The functions Bj(t) represent the temperatures of the three bricks and the function I(t) represents the interior temperature within an adobe house.
Our first step is to define the exterior temperature as shown in the TI-92 screen below.
The programs we are using are fairly powerful and enable us to use fairly big steps with reasonable accuracy. The variable h will be used for the step size. We will use h = 0.50 hours. We will look at a solution over a period of two days or 48 hours. Since each step will be 0.50 hours we need 96 steps.
The last line in the screen above specifies the initial conditions and the number of steps. Notice the syntax -- the number of steps is given first and followed by the initial conditions written as a vector. The program ndsolve determines the number of differential equations by looking at the length of this vector. This line will take some time to evaluate. It will save its results in the variable solution which will be used later to draw a graph.
The differential equations are specified in the function foobar that you have already loaded into your TI-92. You will need to change this function to work with other systems of differential equations. The function foobar is shown in the screen below using the Program Editor.
This program defines five differential equations that describe the derivatives of the variables t, brick1, brick2, brick3 and interior. You may want to look at the TI-92 CAS window for the previous module.
After the program ndsolve in the second TI-92 screen above has finally completed its job you can graph the results as follows. First, use the SEQUENCE graph mode and make sure that the AXES are set to TIME.
Then set up the Y= screen as shown in the TI-92 screens below.
and the WINDOW screen as shown in the TI-92 screen below.
Now, press GRAPH to see the following graph.
This graph shows what happens after 2 days. We need to continue the simulation longer to see what happens over the long term. The following screen shows how to simulate another 2 days. This screen continues calculating using the end result from the previous calculations.
The graph below shows the results. Important: You must go to the Y= screen and re-enter the initial conditions in the form
You will also want to change the WINDOW screen.
The following graph shows the results after an additional two days.
Notice that the graph of the interior temperature (the one with the smallest amplitude) is roughly 12 hours (one-half period) behind the graph of the exterior temperature (the one with the largest amplitude).