Caputo分數階導數

Caputo分數階導數（Caputo fractional derivative），又名Caputo型分數階導數（Caputo-type fractional derivative），是一種非整數階導數的推廣，以Michele Caputo的名字命名。 Caputo於1967年首次定義了該形式的分數階導數。

動機

Caputo分數階導數源自黎曼-劉維爾分數階積分。設${\textstyle f}$在$\left(0,\,\infty \right)$上連續 ，則黎曼-劉維爾分數次積分${\textstyle {^{\text{RL}}\operatorname {I} }}$如下：

$${_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t$$

其中${\textstyle \Gamma \left(\cdot \right)}$是Gamma函數。

定義${\textstyle \operatorname {D} _{x}^{\alpha }:={\frac {\operatorname {d} ^{\alpha }}{\operatorname {d} x^{\alpha }}}}$，滿足${\textstyle \operatorname {D} _{x}^{\alpha }\operatorname {D} _{x}^{\beta }=\operatorname {D} _{x}^{\alpha +\beta }}$，${\textstyle \operatorname {D} _{x}^{\alpha }={^{\text{RL}}\operatorname {I} _{x}^{-\alpha }}}$。若${\textstyle \alpha =m+z\in \mathbb {R} \wedge m\in \mathbb {N} _{0}\wedge 0<z<1}$那麼${\textstyle \operatorname {D} _{x}^{\alpha }=\operatorname {D} _{x}^{m+z}=\operatorname {D} _{x}^{z+m}=\operatorname {D} _{x}^{z-1+1+m}=\operatorname {D} _{x}^{z-1}\operatorname {D} _{x}^{1+m}={^{\text{RL}}\operatorname {I} }_{x}^{1-z}\operatorname {D} _{x}^{1+m}}$ 。故，若$f$亦屬於$C^{m}\left(0,\,\infty \right)$ ， 則有

$${\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.$$

上式稱為Caputo型分數階導數，通常寫為${\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}$ 。

定義

Caputo型分數階導數的首個定義由Caputo給出：

$${^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t$$

其中$C^{m}\left(0,\,\infty \right)$，${\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1}$。

一個常見的等效定義是：

$${^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t$$

其中${\textstyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} }$，${\textstyle \left\lceil \cdot \right\rceil }$是上限函數。通過換元法，令${\textstyle \alpha =m+z}$，則${\textstyle \left\lceil \alpha \right\rceil =m+1}$，${\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1}$，可以得到上述式子。

另一個常見的等效定義如下：

$${^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t$$

其中${\textstyle n-1<\alpha <n\in \mathbb {N} }$ 。

上述定義存在問題：它們只適用於${\textstyle \left(0,\,\infty \right)}$ 。可以通過將積分下限替換為${\textstyle a}$來解決： ${\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}$ 。新的定義域是${\textstyle \left(a,\,\infty \right)}$ .

性質和定理

基本性質和定理

該算子的一些基本性質如下：

基本性質和定理表

特性	$f\left(x\right)$	${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]$	條件
定義	$f\left(x\right)$	$f^{\left(\alpha \right)}\left(x\right)-f^{\left(\alpha \right)}\left(a\right)$
線性	$b\cdot g\left(x\right)+c\cdot h\left(x\right)$	$b\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[g\left(x\right)\right]+c\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[h\left(x\right)\right]$
指數律	$\operatorname {D} _{x}^{\beta }$	${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}$	$\beta \in \mathbb {Z}$
半群性質	${_{a}^{\text{C}}\operatorname {D} _{x}^{\beta }}$	${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}$	$\left\lceil \alpha \right\rceil =\left\lceil \beta \right\rceil$

非交換律

指數律並不總是滿足交換律：

$$\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\operatorname {_{a}^{\text{C}}D} _{x}^{\beta }=\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha +\beta }\neq \operatorname {_{a}^{\text{C}}D} _{x}^{\beta }\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }$$

其中$\alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} \wedge \beta \in \mathbb {N}$ 。

分數萊布尼茨法則

Caputo分數階導數的萊布尼茨法則如下：

$$\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\left[g\left(x\right)\cdot h\left(x\right)\right]=\sum \limits _{k=0}^{\infty }\left[{\binom {a}{k}}\cdot g^{\left(k\right)}\left(x\right)\cdot \operatorname {_{a}^{\text{RL}}D} _{x}^{\alpha -k}\left[h\left(x\right)\right]\right]-{\frac {\left(x-a\right)^{-\alpha }}{\Gamma \left(1-\alpha \right)}}\cdot g\left(a\right)\cdot h\left(a\right)$$

其中${\textstyle {\binom {a}{b}}={\frac {\Gamma \left(a+1\right)}{\Gamma \left(b+1\right)\cdot \Gamma \left(a-b+1\right)}}}$是二項式係數。

與其他分數階微分算子的關係

Caputo型分數階導數的定義與黎曼-劉維爾分數階積分密切相關：

$${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[f\left(x\right)\right]\right]$$

此外，還適用以下關係：

$${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]$$

其中${_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}$是黎曼-劉維爾分數階導數。

拉普拉斯變換

Caputo型分數階導數的拉普拉斯變換如下：

$${\mathcal {L}}_{x}\left\{{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]$$

其中${\textstyle {\mathcal {L}}_{x}\left\{f\left(x\right)\right\}\left(s\right)=F\left(s\right)}$ .

一些函數的Caputo分數階導數

常數$c$的Caputo分數階導數由下式給出：

$${\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}$$

冪函數$x^{b}$的Caputo分數階導數由下式給出：

$${\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}$$

指數函數$e^{a\cdot x}$的Caputo分數階導數由下式給出：

$${\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}$$

其中${\textstyle E_{x}\left(\nu ,\,a\right)={\frac {a^{-\nu }\cdot e^{a\cdot x}\cdot \gamma \left(\nu ,\,a\cdot x\right)}{\Gamma \left(\nu \right)}}}$是${\textstyle \operatorname {E} _{t}}$-函數，${\textstyle \gamma \left(a,\,b\right)}$是下不完全Gamma函數。

拓展閱讀

• Ricardo Almeida, A Caputo fractional derivative of a function with respect to another function