In mathematics, the Grünwald–Letnikov derivative is a basic extension of the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...

that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from

Prague Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and List of cities in the Czech Republic, largest city in the Czech Republic, and the historical capital of Bohemia. On the Vltava river, Prague is home to about 1.3 milli ...

, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in

Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 millio ...

in 1868.

Constructing the Grünwald–Letnikov derivative

The formula :

f'(x) = \lim_ \frac

for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be: :

\beginf''(x)&=\lim_\frac\\&=\lim_\frac\end

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

= \lim_ \frac

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

f^(x) = \lim_ \frac

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

\mathbb^q f(x) = \lim_ \frac\sum_(-1)^m f(x+(q-m)h).

This defines the Grünwald–Letnikov derivative. To simplify notation, we set: :

\Delta^q_h f(x) = \sum_(-1)^m f(x+(q-m)h).

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

\mathbb^q f(x) =  \lim_\frac.

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 :

f^(x) = \lim_ \frac\sum_(-1)^m f(x+mh).

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

\mathbb^q f(x) = \lim_ \frac\sum_(-1)^m f(x+mh).

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: :

\mathbb^q f(x) = \lim_ \frac\sum_(-1)^m f(x-mh).

Constructing the Grünwald–Letnikov derivative

An alternative definition

References

Further reading