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The idea of Grunwald-Letnikov differentiation comes from the following formulas of backward and forward difference equations. Within this entry, $[ \cdot ]$ will be used to denote the greatest integer function and $\Gamma$ will be used to denote the gamma function.
Backward difference
\begin{equation}D_{-}(f)(x) = \lim_{h\to 0}\frac{f(x)-f(x-h)}{h} \end{equation} \begin{equation}D^n_{-}(f)(x)=\lim_{h\to 0}\frac{1}{h^n}\sum_{k=0}^n \frac{(-1)^k n!}{k! (n-k)!}f(x-kh) \end{equation} For derivatives of integer orders, we only requires to specifies one point $x\in {\mathbb R}$ . Fractional derivatives, like fractional definite integrals, require an interval $[a,b]$ to be specified for the function $f:{\mathbb R}\to {\mathbb R}$ we are talking about.
Definition 1: Left-hand Grunwald-Letnikov derivative
\begin{equation}D^p_{-}(f)(x)= \lim_{h\to 0}\frac{1}{h^p}\sum_{k=0}^{\left[\frac{b-a}{h}\right]} \frac{(-1)^k\Gamma (p+1)}{k! \Gamma (p-k+1)} f(x-kh) \end{equation} Forward difference
\begin{equation}D_{+}(f)(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \end{equation} \begin{equation}D^n_{+}(f)(x) = \lim_{h\to 0}\frac{1}{h^n}\sum^n_{k=0} \frac{(-1)^k n!}{k! (n-k)!} f(x+(n-k-1)h) \end{equation} Definition 2: Right-hand Grunwald-Letnikov derivative
\begin{equation}D^p_{+}(f)(x)= \lim_{h\to 0}\frac{1}{h^p}\sum_{k=0}^{\left[\frac{b-a}{h}\right]} \frac{(-1)^k\Gamma (p+1)}{k! \Gamma (p-k+1)} f(x+(m-k-1)h) \end{equation} Theorem 1: Properties of fractional derivatives
- Linearity: $D^p_{\pm}(a f+ b g)(x) = a D^p_{\pm}(f)(x) + b D^p_{\pm}(g)(x)$ where $a,b\in {\mathbb R}$ are any real constants
- Iteration: $D^p_{\pm}D^q_{\pm}(f)(x) = D^{p+q}_{\pm}(f)(x)$
- Chain rule: $\displaystyle{\frac{d^\beta f(g(x))}{dx^{\beta}} =\sum_{k=0}^{\infty}\frac{\Gamma(1+\beta)}{\Gamma(1+k)\Gamma(1-k+\beta)} \frac{d^{\beta-k}1}{dx^{\beta-k}} \frac{d^k f(g(x))}{dx^k} }$
- Leibniz Rule: $\displaystyle{\frac{d^\beta (f(x)g(x))}{dx^\beta} =\sum_{k=0}^{\infty}\frac{\Gamma(1+\beta)}{\Gamma(1+k)\Gamma(1-k+\beta)} \frac{d^k f(x)}{dx^k} \frac{d^{\beta-k}g(x)}{dx^{\beta-k}} }$
Theorem 2: Table of fractional derivatives
- $\displaystyle{ D^{\alpha}_{\pm}(x^p) =\frac{\Gamma (p+1)x^{p-\alpha}}{\Gamma (p-\alpha+1)} }$ where $\alpha,p\in {\mathbb R}$ and $\Gamma(x)$
- $\displaystyle{ D^{\alpha}_{\pm}( e^{\lambda x} ) =\lambda^{\alpha} e^{\lambda x} }$ for all $\lambda\in {\mathbb R}$
- $\displaystyle{ D^{\alpha}_{\pm} (\sin x) = \sin \left(x+\frac{\alpha \pi}{2}\right)}$
- $\displaystyle{ D^{\alpha}_{\pm} (\cos x) = \cos \left(x+\frac{\alpha \pi}{2}\right)}$
- $\displaystyle{ D^{\alpha}_{\pm} (e^{i x}) =\cos \left(x+\frac{\pi\alpha}{2}\right)+i\sin \left(x+\frac{\pi\alpha}{2}\right) }$
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