We can work on Advanced Applied Mathematics

Download and install IODE (https://conf.math.illinois.edu/iode/download.html). Once downloaded and
installed, launch IODE.

  1. When you start the module, it plots two graphs. The upper one shows an odd 2𝜋-periodic
    square wave 𝑓(𝑡). Two periods of this function are shown over a length 4𝜋. It also shows, in red,
    a partial sum
  • ∑(𝑎𝑛 cos 𝑛𝑡 + 𝑏𝑛 sin 𝑛𝑡)
    of the Fourier series. The final terms in this partial sum are cos(𝑁𝑡) and sin(𝑁𝑡), and so IODE
    calls 𝑁 the “top harmonic”. The current value of the top harmonic is displayed in the middle of
    the plotting window, and you can increase or decrease it by clicking on the arrow buttons; doing
    so repeatedly creates an “animation” effect. Or, you can just enter a new top harmonic number
    directly into the box. When you increase the value of the top harmonic, the partial sum should
    better approximate the function.
  1. Now use the Function menu to enter a new function, perhaps 𝑓(𝑡) = |𝑡| (the Matlab code for
    this absolute value function is abs(x)). Try increasing and decreasing top harmonic, to see the
    effect on the partial sums.
  2. The lower graph in the window shows the “error” between 𝑓 and the partial sum of its Fourier
    series, defined just to be the difference
    𝑒𝑟𝑟𝑜𝑟(𝑡) = 𝑓(𝑡) − [
  • ∑(𝑎𝑛 cos 𝑛𝑡 + 𝑏𝑛 sin 𝑛𝑡)
    When we make the top harmonic 𝑁 bigger, we expect the error to get smaller. Try it and see.
    (Note: The vertical scale on the error plot changes, when the error gets smaller, in order to keep
    the error visible.)

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