Grunwald-Letnikov differintegral
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In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly.
It is noted:
- <math>{}_a \mathbb{D}^q_t<math>
and is most generally defined as:
<math>{}_a\mathbb{D}^q_t= \left\{\begin{matrix} \frac{d^q}{dx^q}, & \mathbb{R}(q)>0 \\ 1, & \mathbb{R}(q)=0 \\ \int^t_a(dx)^{-q}, & \mathbb{R}(q)<0. \end{matrix}\right.<math>
The Grunwald-Letnikov differintegral is a commonly used form of the differintegral. It is defined using the definition of the derivative:
- <math>f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}.<math>
Constructing the Grunwald-Letnikov differintegral
The formula 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 symmetrically, this simplifies to:
- <math> = \lim_{h \to 0} \frac{f(x+2h)-2f(x+h)+f(x)}{h^2}<math>
In general, we have (see binomial coefficient):
- <math>d^n f(x) = \lim_{h \to 0} \frac{\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+mh)}{h^n}<math>
If we remove the restriction that n must be a positive integer, we have:
- <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>
This is the Grunwald-Letnikov differintegral.
A simpler expression
We may also write the expression more simply if we make the substitution:
- <math>\Delta^q_h f(x) = \sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+mh)<math>
This results in the expression:
- <math>\mathbb{D}^q f(x) = \lim_{h \to 0}\frac{\Delta^q_h f(x)}{h^q}.<math>