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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 0<\alpha<1 and type 0\leq\mu\leq 1 with respect to x was first introduced in [28, 3, 32]. [401.1.3.2] It is defined by

D^{{\alpha,\mu}}_{{a\pm}}f(x)=\left(\pm I^{{\mu(1-\alpha)}}_{{a\pm}}\frac{\mathrm{d}}{\mathrm{d}x}(I^{{(1-\mu)(1-\alpha)}}_{{a\pm}}f)\right)(x) (7)

for functions for which the expression on the right hand side exists. [401.2.0.1] In this definition the symbols I^{{\alpha}}_{{a\pm}} stand for the (right/left)-sided Riemann-Liouville fractional integral. [401.2.0.2] The right-sided Riemann-Liouville fractional integral of order \alpha>0 is defined for a locally integrable function f on [a,\infty[ as [33]

(I^{{\alpha}}_{{a+}}f)(x)=\frac{1}{\Gamma(\alpha)}\int _{a}^{x}(x-y)^{{\alpha-1}}f(y)\;\mathrm{d}y (8)

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

(I^{{\alpha}}_{{a-}}f)(x)=\frac{1}{\Gamma(\alpha)}\int _{x}^{a}(y-x)^{{\alpha-1}}f(y)\;\mathrm{d}y (9)

for x<a. [401.2.0.3] The Riemann-Liouville fractional derivative corresponds to the special case \mu=0. [401.2.0.4] It is the most frequently used definition of a fractional derivative. [401.2.0.5] The special case \mu=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<\alpha<1[3]

\mathscr L\{ D^{{\alpha,\mu}}_{{a+}}f(x)\}(u)=u^{\alpha}\mathscr L\{ f(x)\}(u)-u^{{\mu(\alpha-1)}}(I^{{(1-\mu)(1-\alpha)}}_{{a+}}f)(0+) (10)

where the initial value (I^{{(1-\mu)(1-\alpha)}}_{{a+}}f)(0+) is the Riemann-Liouville integral of order (1-\mu)(1-\alpha) evaluated in the limit t\to 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 \mu=1 involve f(0+) as initial value.