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The Fractional Derivative

April 8th, 2009 |

Author: Merrin

The formula for the fractional integral leads to the formula for the fractional derivative. The fractional derivative of order q can be written as an integer derivative of order n with a fractional integration of order nu. The number n is the smallest integer greater than q and nu is the difference between n and q.

$q = n- \nu \\ \\{}_{a} D_{x}^{q} f(x)={}_{a} D_{x}^{n} {}_{a} D_{x}^{-\nu } f(x)$

Another acceptable notation is to write

$\cfrac{d^{q} }{[d(x-a)]^{q} } f(x)=\cfrac{d^{n} }{[d(x-a)]^{n} } \cfrac{d^{-\nu } }{[d(x-a)]^{-\nu } } f(x)$

The nature of the fractional derivative therefore having limits resolves the paradox between our earlier notions of fractional differentiation exponential functions and polynomials. The lower limit for the fractional differentiation of the polynomials was zero, while the lower limit for the fractional differentiation of the exponential functions was negative infinity. When the lower limit of the derivative is negative infinity, such a derivative is said to be a Weyl derivative. Most of the main results of fractional calculus though are with a different lower limit to keep the results finite so the Weyl derivative is generally only used in special cases involving exponentials.

Another approach to calculating the fractional derivative is to extend the definition of the nth order derivative to fractional orders.

$\begin{array}{l}{\displaystyle D^{n} f(x)=\mathop{\lim }\limits_{\Delta x\to 0} \left(\Delta x\right)^{-n} \sum _{j=0}^{n}(-1)^{j} \frac{n!}{j!\left(n-j\right)!} f\left(x-j\Delta x\right)} \\{\displaystyle D^{q} f(x)=\mathop{\lim }\limits_{N\to \infty} \left(\frac{x-a}{N} \right)^{-q} \sum _{j=0}^{N-1}(-1)^{j} \frac{q!}{j!\left(q-j\right)!} f\left(x-j\frac{x-a}{N} \right)} \end{array}$

One key observation is that you can extend the upper limit of the summation to infinity for integer orders because the binomial coefficients cut off. Calculations with this formula however are notoriously difficult except for the simplest situations.

Now we have an updated impression of the fractional derivative in terms of modern theory. Fractional derivatives are the other half of the picture. We know how to express them in several different equivalent forms, but calculating them is a matter of calculus.

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