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The L2 method applied to the Caputo derivative with $\alpha = 1.5$ looks like
h
Error
EOC
3.333e-02
6.659753e-02
---
1.613e-02
3.341454e-02
0.950
7.937e-03
1.690615e-02
0.961
3.937e-03
8.552690e-03
0.972
1.961e-03
4.318041e-03
0.980
9.785e-04
2.175312e-03
0.986
6.519e-04
1.455290e-03
0.990
4.888e-04
1.093799e-03
0.991
Overall
0.975
The L2 method applied to the Riemann-Liouville derivative with $\alpha = 1.5$ (through D_{RL} = D_C - F(a)) gives
h
Error
EOC
3.333e-02
3.686930e-02
---
1.613e-02
9.678625e-03
1.842
7.937e-03
2.593723e-03
1.857
3.937e-03
6.847805e-04
1.900
1.961e-03
1.774681e-04
1.937
9.785e-04
4.536133e-05
1.962
6.519e-04
2.033538e-05
1.976
4.888e-04
1.149314e-05
1.981
Overall
1.916
For the Caputo derivative, it's first-order regardless of $\alpha$, but for the RL derivative it seems to behave roughly like $\alpha + 0.5$ instead of the expected $3 - \alpha$. Note quite sure what to make of that.
The L2C method seems to be working correctly. Likely just an index issue or something? Needs a bit of debugging.
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