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$f$在$\left( 0,\, \infty \right)$上連續，則黎曼-劉維爾分數次積分${^{\text{RL}}\operatorname{I}}$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$\Gamma\left( \cdot \right)$是Gamma函數。

定義$\operatorname{D}_{x}^{\alpha} := \frac{\operatorname{d}^{\alpha}}{\operatorname{d}x^{\alpha}}$，滿足$\operatorname{D}_{x}^{\alpha} \operatorname{D}_{x}^{\beta} = \operatorname{D}_{x}^{\alpha + \beta}$，$\operatorname{D}_{x}^{\alpha} = {^{\text{RL}}\operatorname{I}_{x}^{-\alpha}}$。若$\alpha = m + z \in \mathbb{R} \wedge m \in \mathbb{N}_{0} \wedge 0 < z < 1$那麼$\operatorname{D}_{x}^{\alpha} = \operatorname{D}_{x}^{m + z} = \operatorname{D}_{x}^{z + m} = \operatorname{D}_{x}^{z - 1 + 1 + m} = \operatorname{D}_{x}^{z - 1}\operatorname{D}_{x}^{1 + m} = {^{\text{RL}}\operatorname{I}}_{x}^{1 - z}\operatorname{D}_{x}^{1 + m}$。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${ ^{\text{C}}\operatorname{D}}_{x}^{\alpha}$。

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

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$\alpha \in \mathbb{R}_{> 0} \setminus \mathbb{N}$，$\left\lceil \cdot \right\rceil$是上限函數。通過換元法，令$\alpha = m + z$，則$\left\lceil \alpha \right\rceil = m + 1$，$\left\lceil \alpha \right\rceil + z = \alpha + 1$，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$n - 1 < \alpha < n \in \mathbb{N}$。

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

## 性質和定理

### 基本性質和定理

該算子的一些基本性質如下：

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：

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

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$\binom{a}{b} = \frac{\Gamma\left( a + 1 \right)}{\Gamma\left( b + 1 \right) \cdot \Gamma\left( a - b + 1 \right)}$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$\mathcal{L}_{x}\left\{ f\left( x \right) \right\}\left( s \right) = F\left( s \right)$ .

## 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：

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

冪函數$x^{b}$的Caputo分數階導數由下式給出：

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

指數函數$e^{a \cdot x}$的Caputo分數階導數由下式給出：

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

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