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
   𝑎0
   2

• ∑(𝑎𝑛 cos 𝑛𝑡 + 𝑏𝑛 sin 𝑛𝑡)
  𝑁
  𝑛=1
  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) − [
   𝑎0
   2

• ∑(𝑎𝑛 cos 𝑛𝑡 + 𝑏𝑛 sin 𝑛𝑡)
  𝑁
  𝑛=1
  ]
  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.)

Sample Solution