[123.6.1] The theory of fractional
dynamics yields a three parameter function,
that allows to fit both, the asymmetric peak and
the excess wing with a single stretching exponent
[17, 16, 23].
[123.6.2] The three parameter function is denoted as
“fractional dynamics” (FD) relaxation in
figure 2.
[123.6.3] Its functional form reads
| ft | =E1,1-α,1-tτ1,-τ2ατ1t1-α | | | (8a) |
| εu | =1+uτ2α1+uτ2α+uτ1 | (FD). | | (8b) |
where
| Ea1,a2,bz1,z2=∑k=0∞∑ℓ1≥0∑ℓ2≥0ℓ1+ℓ2=kk!ℓ1!ℓ2!z1ℓ1z2ℓ2Γb+a1ℓ1+a2ℓ2 | | (9) |
with a1,a2>0 and b,z1,z2∈C
is the binomial Mittag-Leffler function [25].
[123.6.4] The function ft from eq. (8)
solves the fractional differential equation
| τ1dfdt+τ2αAαf=-f | | (10) |
with τ1,τ2>0, 0<α<1, and inital value f0=1
[25].
[123.6.5] In eq. (10)
the operator Aα is the infinitesimal generator of
fractional time evolutions of index α
[7, 11, 10, 9, 15, 23].
[123.6.6] It can be written as a fractional time derivative of
order α in the form
where A=-d/dt is the infinitesimal generator of translations.
[123.6.7] For a mathematical definition of Aα see [23].
[123.6.8] Note, that the solution (8) of eq. (10)
holds also for generalized Riemann-Liouville operators
Aα=D0+α,γ of order α and type 0<γ<1
as shown in [17, 16].
[123.6.9] The (right-/left-sided) generalized Riemann-Liouville fractional
derivative of order 0<α<1 and type
0≤γ≤1 with respect to x was introduced in definition
3.3 in [15, p.113] by
| Da±α,γfx=±Ia±γ1-αddxIa±1-γ1-αfx | | (12) |
where
| Ia+αfx=1Γα∫axx-yα-1fydy | | (13) |
for x>a,
denotes the right-sided
Riemann-Liouville fractional integral of order α>0,
and the left sided integral Ia-α is defined analogously
[15, 13].
