不同分数阶导数程序

以下是不同分数阶导数的程序实现：

1. 一阶导数：

import numpy as np

def first_derivative(f, h):
    n = len(f)
    df = np.zeros(n)

    for i in range(1, n-1):
        df[i] = (f[i+1] - f[i-1]) / (2*h)

    df[0] = (f[1] - f[0]) / h
    df[n-1] = (f[n-1] - f[n-2]) / h

    return df

2. 二阶导数：

import numpy as np

def second_derivative(f, h):
    n = len(f)
    d2f = np.zeros(n)

    for i in range(1, n-1):
        d2f[i] = (f[i+1] - 2*f[i] + f[i-1]) / (h**2)

    d2f[0] = (f[2] - 2*f[1] + f[0]) / (h**2)
    d2f[n-1] = (f[n-3] - 2*f[n-2] + f[n-1]) / (h**2)

    return d2f

3. 分数阶导数：

import numpy as np
from scipy.special import gamma

def frac_derivative(f, alpha, h):
    n = len(f)
    df = np.zeros(n)

    for i in range(1, n-1):
        df[i] = sum([(f[i+j+1] - f[i+j-1]) / (2*h) for j in range(-i, i)]) / (h**alpha * gamma(alpha+1))

    df[0] = sum([(f[i+1] - f[i-1]) / (2*h) for i in range(alpha)]) / (h**alpha * gamma(alpha+1))
    df[n-1] = sum([(f[n-1-i-1] - f[n-1-i+1]) / (2*h) for i in range(alpha)]) / (h**alpha * gamma(alpha+1))

    return df

其中，alpha 表示分数阶导数的阶数，gamma 函数为阶乘的推广。