chalkdust the Riemann–Liouville derivative is D9¾ f (x) =
9! −1/4 x , Γ( 14 )
which is clearly diﬀerent to the Caputo derivative. Two things are to be noted here. The fractional part is only contained in the integral, so in order to obtain both of the 9¾ derivatives of a function we need to quarter integrate the tenth derivative or diﬀerentiate the ¼ integral ten times. Also, and very importantly in fractional calculus, the fractional integral depends on its integration limits (just as in the traditional version of calculus) but since the fractional derivative is defined in terms of the fractional integral, then the fractional derivatives also depend on the limits. There are many applications of fractional calculus in, for example, engineering and physics. Interestingly, most of the applications have emerged in the last twenty years or so, and it has allowed a diﬀerent approach to topics such as viscoelastic damping, chaotic systems and even acoustic wave propagation in biological tissue. Perhaps fractional calculus is a bit tricky to interpret, seeming at first to be a weird generalisation of calculus but for me, just thinking about the 9¾ derivative of a function was like discovering the entry into a whole new world between platforms 9 and 10. Certainly, there is some magic hidden behind fractional calculus!
The fractional derivatives of Dν x 9 , with ν between 0 and 10, and x between 0 and 1.
Rafael Prieto Curiel is doing a PhD in mathematics and crime. You can follow him on Twier @rafaelprietoc or visit his blog rafaelprietoc.wordpress.com
Chessboard Squares It was once claimed that there are 204 squares on a chessboard. Can you justify this claim? Source: Thinking Mathematically by John Mason, Leone Burton & Kaye Stacey Answers at chalkdustmagazine.com