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Caputo分数阶导数

Wikipedia · 2026-07-13 维基百科 👁 1 阅读

Caputo分数阶导数（Caputo fractional derivative），又名Caputo型分数阶导数（Caputo-type fractional derivative），是一种非整数阶导数的推广，以Michele Caputo的名字命名。 Caputo于1967年首次定义了该形式的分数阶导数。

动机
Caputo分数阶导数源自黎曼-刘维尔分数阶积分。设f在\left( 0,\, \infty \right)上连续 ，则黎曼-刘维尔分数次积分{^{\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函数。

定义\operatorname{D}_{x}^{\alpha} := \frac{\operatorname{d}^{\alpha}}{\operatorname{d}x^{\alpha}}，满足\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 那么\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亦属于C^{m}\left( 0,\, \infty \right) ， 则有

{\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.

上式称为Caputo型分数阶导数，通常写为{ ^{\text{C}}\operatorname{D}}_{x}^{\alpha} 。

定义
Caputo型分数阶导数的首个定义由Caputo给出：

{^{\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 。

一个常见的等效定义是：

{^{\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是上限函数。通过换元法，令\alpha = m + z，则\left\lceil \alpha \right\rceil = m + 1，\left\lceil \alpha \right\rceil + z = \alpha + 1 ，可以得到上述式子。

另一个常见的等效定义如下：

{^{\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 。

上述定义存在问题：它们只适用于\left( 0,\, \infty \right) 。可以通过将积分下限替换为a来解决： {_{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 。新的定义域是\left( a,\, \infty \right) .

性质和定理
基本性质和定理
该算子的一些基本性质如下：

非交换律
指数律并不总是满足交换律：

\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} 。

分数莱布尼茨法则
Caputo分数阶导数的莱布尼茨法则如下：

\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)}是二项式系数。

与其他分数阶微分算子的关系
Caputo型分数阶导数的定义与黎曼-刘维尔分数阶积分密切相关：

{_{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，还适用以下关系：

{_{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-刘维尔分数阶导数。

拉普拉斯变换
Caputo型分数阶导数的拉普拉斯变换如下：

\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) .

一些函数的Caputo分数阶导数
常数c的Caputo分数阶导数由下式给出：

\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}

幂函数x^{b}的Caputo分数阶导数由下式给出：

\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

指数函数e^{a \cdot x}的Caputo分数阶导数由下式给出：

\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}-函数，\gamma \left( a,\, b \right)是下不完全Gamma函数。

参考文献
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