8.03 at ESG - Notes

Fourier Series using MAPLE

Online help is available for both Maple and Matlab:

• If you need an introduction to Maple, check out Using Maple for ESG Subjects.
• Maple at MIT.
• Maple on Athena.
• Athena Consulting Maple Stock Answers.
• MATLAB at MIT.
• What Runs Where on Athena and What's New are good to have handy.

• Release 3 worksheet
• Release 3 ASCII commands
• Release 8 worksheet (to download .mws files, it's "Shift-Left")
• Release 8 ASCII commands

To start Maple from an Athena terminal, either start from the dash, or, at the Athena prompt, do

athena% add maple
athena% maple&

Now, wait. Maple, like MATLAB, takes a while to load. Also like MATLAB, the older versions load and run faster (go figure). To load an older release, the command is
athena% maple -ver 5r3&
(the commands given here will run on the either release, with the single exception noted).
If you've downloaded the worksheet, or have your own, you open these from the File menu item.

Start by telling Maple what you think you want to do. Specifically, enter

assume(n,integer);

This tells Maple that the index n you'll be using is to be taken as an integer.
Next, define the function you want to analyze. In this case, there will be two pieces, one for 0 < z < d and one for d < z < L. One way to do this in Maple is

f1:=z*a/d; f2:=(L-z)*a/(L-d);

Note that the applicable ranges are not given; this will be done when the command to do the integral is invoked. Here goes:

B(n):=int(f1*sin(n*Pi*z/L),z=0..d)+int(f2*sin(n*Pi*z/L),z=d..L);

In the above, note that the range of z is indeed broken up into two intervals, and the integral that defines B(n) is the sum of these integrals.
(These notes use B(n) instead of the text's A_m for calculational and historic purposes; convince yourself that it doesn't matter.)

What you have may look like a mess. To make it look nicer, enter the command

simplify("); in Release 3, or

simplify(%); in Release 8.

Compare this to the expression for A_m given in B&B Problem 2.11.

In order to see how these coefficients will indeed reproduce the given initial condition s(z, 0), we need to give specific values for A, d and L. One such choice is

L:=7*d; a:=1; d:=1;

To get the sum of the first, say, 25 terms, the command is

S2:=(2/L)sum(B(n)*sin(n*Pi*x), n=1..25):

In the above, a colon instead of a semicolon was used to end the line; this suppresses the display of a rather long expression. Again, S2 for the sum is for historic purposes; any variable not used so far (but not D, E or I, which are reserved for special math purposes) could be used. Also note that the normalization factor (2/L) has been included at this stage.

The punchline is the plot of S2:

plot(S2,x=0..1);

Lastly, for now, it is possible to display the individual terms, but essentially the B(n) need to be recalculated; this is really no big deal. In the given example, one possible set of commands would be:

S3:=plot(S2,x=0..1,color=magenta):
with(plots):
S4:={(2/L)*(int(f1*sin(i*Pi*z/L),z=0..d)+int(f2*sin(i*Pi*z/L),z=d..L)) *sin(i*Pi*x) $ i=1..5}:
S5:=plot(S4,x=0..1): display([S5,S3]);

In the above, the plot formerly called S2 has been renamed and given a color, neither of which is necessary. S4 is a set of functions, iterated with the $ i=1..5 notation, and S5 is the plot of this set of functions. The "display" allows simultaneous presentation of plots generated by different commands.
For long commands, do not use returns, as this will enter the command whether or not you've completed it "Shift-Return" is a discretionary linebreak within commands.

There are many things beyond those presented here that can be done; the plot output could be varied to a great extent, including line thickness and furthe options for colors. Such details will not be addressed here (see the help menus).