3 Derivatives of non-integer order and non-integer type

[401.1.2.1] There are many definitions for derivatives of non-integer order (see [28] for a recent introduction). [401.1.2.2] A new one-parameter family of Riemann-Liouville type derivatives was introduced in [3]. [401.1.2.3] Its definition will now be repeated.

[401.1.3.1] The (right-/left-sided) fractional derivative of order 0<α<1 and type 0≤μ≤1 with respect to x was first introduced in [28, 3, 32]. [401.1.3.2] It is defined by

for functions for which the expression on the right hand side exists. [401.2.0.1] In this definition the symbols Ia±α stand for the (right/left)-sided Riemann-Liouville fractional integral. [401.2.0.2] The right-sided Riemann-Liouville fractional integral of order α>0 is defined for a locally integrable function f on [a,∞[ as [33]

for x>a, the left-sided Riemann-Liouville fractional integral is defined as

for x<a. [401.2.0.3] The Riemann-Liouville fractional derivative corresponds to the special case μ=0. [401.2.0.4] It is the most frequently used definition of a fractional derivative. [401.2.0.5] The special case μ=1 is sometimes called Caputo fractional derivative [34, 35], others attribute it to Liouville [33].

[401.2.1.1] The difference between fractional derivatives of different types becomes apparent from Laplace transformation. [401.2.1.2] One finds for 0<α<1[3]

where the initial value Ia+1-μ⁢1-α⁢f⁢0+ is the Riemann-Liouville integral of order 1-μ⁢1-α evaluated in the limit t→0+. [401.2.1.3] This shows that the type of the fractional derivative determines the initial values to be used in applications, resp. the initial values determine the type of derivative to be used. [401.2.1.4] Note that not only derivatives of integer order but also fractional derivatives of type μ=1 involve f⁢0+ as initial value.