Comments (1)
Hello @CheapMeow, the asymptotic behavior at the left endpoint is a side effect of how the Grunwald-Letnikov differintegral is implemented. Comparing to the result gained from the Riemann-Liouville differintegral and the expected results, we see that it much more accurate.
These results were obtained using the following code:
import numpy as np
import differint.differint as df
import matplotlib.pyplot as plt
def f(t):
return np.power(t, 3)
def neg_f(t):
return f(-1 * t)
alpha = 0.6
t_2 = np.linspace(0, 1, 100)
GL_2 = df.GL(alpha, f, 0, 1, 100)
GL_2_rev = df.RL(alpha, f, 0, 1, 100)
expected = lambda t: 6/(df.Gamma(3-alpha+1)) * t ** (3 - alpha)
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.plot(t_2, GL_2, label='GL')
ax.plot(t_2, GL_2_rev, marker='o', markevery=5, label='RL')
ax.plot(t_1, expected(t_1), label='expected')
ax.legend()
plt.show()
Another thing to point out, is that to conserve many of the properties one might expect from the fractional derivative of polynomials, the left endpoint should be kept to 0; I only included that case.
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