And here is a Maple worksheet I will also use: modes_cleaner.mws, modes_cleaner.pdf.
(b) [3 POINTS] Recalling that complex roots correspond to oscillation, and
negative real part corresponds to decay, while positive real part
corresponds to growth, describe or sketch what you'd see if your initial condition
was (i) pure mode n=1, (ii) pure mode n=7, (iii) pure mode n=12.
(c) [2 POINTS] Describe and explain, in light of results in (a) and (b), what you observe
with the applet for the parameter choice above.
(d) [3 POINTS] We've analyzed how and when "stripes" can arise in a
raction-diffusion system, and we've observed a qualitatively
similar phenomenon in wind-formed ripples on sand.
Can you think of any other examples in the world or universe
where stripes form spontaneously like this. Email me pictures
(or URLs of pictures) if you can (Subject line: 455 stripes).
2. [5 POINTS] Find a linear combination of X(x-h), X(x), and X(x+h) that
equals X''(x) + an error which is O(h), i.e. which goes to zero
as h goes to zero no slower than a multiple of h. Basically, the
job here is to pick the multiples so as to leave X''(x) and
cancel out as many other terms as possible. What you are finding
is called a finite-difference approximation to the second derivative.
3. [1 POINT FOR EACH DISTINCT STATE: ABOVE 4 IS EXTRA CREDIT]
How many distinct non-transient states (spatial, temporal, or both)
can you find in the Brusselator at D1=0.0016, D2=0.0060, A=1.5, B=4.6?
Describe each one, perhaps using a sketch of the space-time picture.
You can kick the system around by changing a parameter and then
changing it back again. Note that you have to hit the Enter key
to get a changed parameter value to register. Prize for student with the largest number of states.
PROBLEM 4 NOW MOVED TO HW#8: 4. Replicating the Pearson experiment. I have implemented in C++ the numerical solution of the 2D reaction-diffusion equations for the Gray-Scott model using the method and conditions described in John Pearson's paper. Here is the code: pearson.c.pdf. Now I assume most of you do NOT know C++, but I am hoping that the important bits will be largely comprehensible to you anyway. (a) Make as many specific line-to-line correspondences as you can between the line-numbered version of the Pearson paper: p1, p2, and the line-numbered C++ code. For each correspondence give the line(s) in the paper, the line number(s) in the code, and a brief note about the correspondence. You will get points for each correspondence correctly identified and described. (b) Running the code. In one of the Linux Labs (e.g. the one in Bell Hall), or on your own Linux/Intel machine, make a directory to work in by typing mkdir pearson in a terminal. (Get a terminal by Penguin -> System Tools -> Terminal.) Then download the compiled version of the code: pearson.exe into that directory. Move into that directory by typing cd pearson Make the file runnable by typing chmod +x pearson.exe Then you can run it by typing ./pearson.exe You will be prompted for a file name prefix (best if you use your own last name with no spaces), parameter values, and the times you want snapshots at. Enter your own values, as assigned in class, and let the program run. (Try a short run first, to make sure everything is working as expected.) Check that your snapshots are satisfactory by looking at them with a browser, as described by the program when it ends. Finally, when you are satisfied with the results of your experiment, package and mail them to me PRECISELY as follows. Make a package of all the files you want to send by typing tar cvf yourname.tar *.html *.png, or if you don't want to send ALL the html and png files in the directory, more selectively, something like tar cvf yourname.tar ringland_expt1.html ringland_expt2.html ringland_expt1*.png ringland_expt2*.png Then send me the file "yourname.tar" as an attachment to an email with the Subject line "455 reaction-diffusion experiment". You could, for example, use UB Webmail from within the Mozilla browser. All the results of experiments that I've received so far are at http://orange.math.buffalo.edu//455/pearson_runs_students/ .