Table 2.
Comparing the approximation error for the fractional derivative of polynomials when using the midpoint (MP) rule versus the Prony series. Here Δt = 0 denotes the norm, ∥ε∥0, of the truncation error predicted in Lemma 1.
| Δt | MP | GL | Prony | Birk and Song | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| N = 3 | N = 6 | N = 9 | N = 12 | N = 3 | N = 6 | N = 9 | N = 12 | |||||
| α = 0.1 | 1e–1 | 1.84e0 | 5.24e–1 | 5.07e–1 | 5.14e–1 | 5.14e–1 | 5.14e–1 | 4.73e–1 | 4.95e–1 | 5.12e–1 | 5.14e–1 | |
| 1e–2 | 2.64e–1 | 2.86e–2 | 4.96e–2 | 1.37e–2 | 1.49e–2 | 1.56e–2 | 2.99e–1 | 6.78e–2 | 2.77e–2 | 1.84e–2 | ||
| 1e–3 | 2.76e–2 | 6.64e–3 | 5.04e–2 | 2.76e–3 | 4.45e–4 | 3.81e–4 | 3.01e–1 | 7.42e–2 | 2.23e–2 | 7.31e–3 | ||
| 1e–4 | 2.81e–3 | 2.35e–3 | 5.04e–2 | 2.74e–3 | 3.03e–4 | 4.47e–5 | 3.01e–1 | 7.46e–2 | 2.24e–2 | 6.81e–3 | ||
| 1e–5 | 2.86e–4 | 9.19e–4 | 5.04e–2 | 2.75e–3 | 3.13e–4 | 6.39e–5 | 3.01e–1 | 7.46e–2 | 2.24e–2 | 6.78e–3 | ||
| 1e–6 | 2.93e–5 | 3.92e–4 | 5.04e–2 | 2.75e–3 | 3.14e–4 | 6.69e–5 | 3.01e–1 | 7.46e–2 | 2.24e–2 | 6.77e–3 | ||
| 0 | – | – | 5.04e–2 | 2.75e–3 | 3.14e–4 | 6.71e–5 | 3.01e–1 | 7.46e–2 | 2.24e–2 | 6.77e–3 | ||
| α = 0.4 | 1e–1 | 4.12e0 | 1.80e0 | 1.95e0 | 1.91e0 | 1.91e0 | 1.92e0 | 2.73e0 | 2.22e0 | 1.99e0 | 1.94e0 | |
| 1e–2 | 9.12e–1 | 2.36e–1 | 2.88e–1 | 2.25e–1 | 2.48e–1 | 2.58e–1 | 1.23e0 | 5.25e–1 | 3.34e–1 | 3.22e–1 | ||
| 1e–3 | 1.43e–1 | 9.34e–2 | 2.21e–1 | 1.38e–2 | 1.08e–2 | 1.35e–2 | 1.21e0 | 4.74e–1 | 1.63e–1 | 7.59e–2 | ||
| 1e–4 | 2.53e–2 | 6.46e–2 | 2.21e–1 | 1.14e–2 | 2.04e–3 | 9.88e–4 | 1.21e0 | 4.17e–1 | 1.43e–1 | 4.39e–2 | ||
| 1e–5 | 5.27e–3 | 5.05e–2 | 2.21e–1 | 1.17e–2 | 2.76e–3 | 1.53e–3 | 1.21e0 | 4.12e–1 | 1.38e–1 | 4.31e–2 | ||
| 1e–6 | 1.22e–3 | 4.42e–2 | 2.21e–1 | 1.18e–2 | 2.90e–3 | 1.68e–3 | 1.21e0 | 4.12e–1 | 1.37e–1 | 4.34e–2 | ||
| 0 | – | – | 2.21e–1 | 1.18e–2 | 2.92e–3 | 1.69e–3 | 1.21e0 | 4.12e–1 | 1.37e–1 | 4.35e–2 | ||
| α = 0.8 | 1e–1 | 1.13e1 | 8.46e0 | 8.14e0 | 8.37e0 | 8.42e0 | 8.47e0 | 8.22e0 | 9.10e0 | 9.38e0 | 9.36e0 | |
| 1e–2 | 6.42e0 | 1.84e0 | 2.27e0 | 2.79e0 | 2.92e0 | 3.06e0 | 4.54e0 | 5.35e0 | 5.90e0 | 5.53e0 | ||
| 1e–3 | 3.54e0 | 5.66e–1 | 7.41e–1 | 3.52e–1 | 3.86e–1 | 4.71e–1 | 4.48e0 | 2.80e0 | 3.19e0 | 3.70e0 | ||
| 1e–4 | 2.16e0 | 6.11e–1 | 6.75e–1 | 6.67e–2 | 6.50e–2 | 6.62e–2 | 3.98e0 | 1.71e0 | 2.41e0 | 2.07e0 | ||
| 1e–5 | 1.36e0 | 1.14e0 | 6.75e–1 | 4.34e–2 | 2.98e–2 | 2.29e–2 | 2.85e0 | 1.70e0 | 1.13e0 | 1.42e0 | ||
| 1e–6 | 8.56e–1 | 2.58e0 | 6.75e–1 | 4.90e–2 | 3.54e–2 | 2.76e–2 | 2.89e0 | 1.25e0 | 1.10e0 | 7.39e–1 | ||
| 0 | – | – | 6.75e–1 | 4.98e–2 | 3.64e–2 | 2.87e–2 | 2.89e0 | 8.03e–1 | 2.63e–1 | 8.89e–2 | ||