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Riemann--Liouville
Riemann--Liouville derivatives are fundamental analytically but use different initial-data structures.
Fractional memory
Fractional derivatives include history through singular or weakly singular kernels.
Caputo
Caputo derivatives allow classical initial conditions and are common in applications.
Short memory
Short-memory acceleration reduces cost but must be validated against full-history results.
Mittag--Leffler
Mittag--Leffler functions generalize exponentials in many fractional linear systems.
L1 scheme
The L1 scheme approximates Caputo derivatives through a history convolution.