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 Vasilievich Letnikov (1837–1888) in Moscow in 1868.
The formula
f'(x)=\limh
f(x+h)-f(x) | |
h |
\begin{align}f''(x)&=\limh\to0
f'(x+h)-f'(x) | |
h |
\\&=\lim | |
h1\to0 |
| |||||||
h2 |
-\lim\limits | |
h2\to0 |
\dfrac{f(x+h2)-f(x)}{h2}}{h1}\end{align}
Assuming that the h 's converge synchronously, this simplifies to:
=\limh
f(x+2h)-2f(x+h)+f(x) | |
h2 |
which can be justified rigorously by the mean value theorem. In general, we have (see binomial coefficient):
f(n)(x)=\limh
\sum\limits0\le(-1)m{n\choosem | |
f(x+(n-m)h)}{h |
n}
Removing the restriction that n be a positive integer, it is reasonable to define:
Dqf(x)=\limh
1 | |
hq |
\sum0(-1)m{q\choosem}f(x+(q-m)h).
This defines the Grünwald–Letnikov derivative.
To simplify notation, we set:
q | |
\Delta | |
h |
f(x)=\sum0(-1)m{q\choosem}f(x+(q-m)h).
So the Grünwald–Letnikov derivative may be succinctly written as:
Dqf(x)=\limh
| ||||||||||
hq |
.
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(n)(x)=\limh
(-1)n | |
hn |
\sum0(-1)m{n\choosem}f(x+mh).
or removing the restriction that n must be a positive integer:
Dqf(x)=\limh
(-1)q | |
hq |
\sum0(-1)m{q\choosem}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:
Dqf(x)=\limh
1 | |
hq |
\sum0(-1)m{q\choosem}f(x-mh).