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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 and type with respect to was first introduced in [28, 3, 32]. [401.1.3.2] It is defined by

 (7)

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

 (8)

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

 (9)

for . [401.2.0.3] The Riemann-Liouville fractional derivative corresponds to the special case . [401.2.0.4] It is the most frequently used definition of a fractional derivative. [401.2.0.5] The special case 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 [3]

 (10)

where the initial value is the Riemann-Liouville integral of order evaluated in the limit . [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 involve as initial value.