View source for Grünwald–Letnikov derivative

{{Refimprove|date=June 2009}}
In [[mathematics]], the '''Grünwald–Letnikov derivative''' is a basic extension of the [[derivative]] in [[fractional calculus]] that allows one to take the derivative a non-integer number of times. It was introduced by [[Anton Karl Grünwald]] (1838–1920) from [[Prague]], in 1867, and by [[Aleksey Letnikov|Aleksey Vasilievich Letnikov]] (1837–1888) in [[Moscow]] in 1868.

==Constructing the Grünwald–Letnikov derivative==

The formula 
:<math>f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}</math>
for the derivative can be applied recursively to get higher-order derivatives.  For example, the second-order derivative would be:

:<math>f''(x) = \lim_{h \to 0} \frac{f'(x+h)-f'(x)}{h}</math>

:<math> = \lim_{h_1 \to 0} \frac{\lim_{h_2 \to 0} \frac{f(x+h_1+h_2)-f(x+h_1)}{h_2}-\lim_{h_2 \to 0} \frac{f(x+h_2)-f(x)}{h_2}}{h_1}</math>

Assuming that the ''h'' 's converge synchronously, this simplifies to:

:<math> = \lim_{h \to 0} \frac{f(x+2h)-2f(x+h)+f(x)}{h^2},</math>

which can be justified rigorously by the [[mean value theorem]]. In general, we have (see [[binomial coefficient]]):

:<math>f^{(n)}(x) = \lim_{h \to 0} \frac{\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+(n-m)h)}{h^n}.</math>

Removing the restriction that ''n'' be a positive integer, it is reasonable to define:

:<math>\mathbb{D}^q f(x) = \lim_{h \to 0} \frac{1}{h^q}\sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+(q-m)h).</math>

This defines the Grünwald–Letnikov derivative.

To simplify notation, we set:

:<math>\Delta^q_h f(x) = \sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+(q-m)h).</math>

So the Grünwald–Letnikov derivative may be succinctly written as:

:<math>\mathbb{D}^q f(x) =  \lim_{h \to 0}\frac{\Delta^q_h f(x)}{h^q}.</math>

===An alternative definition===

In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as

:<math>f^{(n)}(x) = \lim_{h \to 0} \frac{(-1)^n}{h^n}\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+mh).</math>

or removing the restriction that ''n'' must be a positive integer:

:<math>\mathbb{D}^q f(x) = \lim_{h \to 0} \frac{(-1)^q}{h^q}\sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+mh).</math>

This equation is called the reverse Grünwald–Letnikov derivative. If the substitution ''h'' → −''h'' is made, the resulting equation is called the direct Grünwald–Letnikov derivative:<ref>http://www.diogenes.bg/fcaa/volume7/fcaa74/74_Ortigueira_Coito.pdf</ref>

:<math>\mathbb{D}^q f(x) = \lim_{h \to 0} \frac{1}{h^q}\sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x-mh).</math>

== References ==

<references />

* ''The Fractional Calculus'', by Oldham, K.; and Spanier, J. Hardcover: 234 pages. Publisher: Academic Press, 1974. {{ISBN|0-12-525550-0}}
* ''From Differences to Derivatives'', by Ortigueira, M. D., and F. Coito.  Fractional Calculus and Applied Analysis 7(4). (2004): 459-71.

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[[Category:Fractional calculus]]