7 Composite fractional relaxation

[404.2.1.1] In this section the general procedure of replacing time translations with composite fractional translations is applied to the simple relaxation equation (19). [404.2.1.2] Proceeding along the same lines as in Section 5 and introducing the fractional derivatives of order α and type μ into the infinitesimal generator A~α one arrives at the composite fractional relaxation equation in the form

with two relaxation times 0<τ1,τ2<∞ and initial condition f⌃⁢0+=1 as before.

[404.2.2.1] A first advantage of the replacement T1⁢t→T~α⁢t over the replacement T1⁢t→Tα⁢t emerges when eq. (32) is Laplace transformed. [404.2.2.2] Using eq. (10) one finds

[404.2.2.3] If the normalized relaxation function f⌃⁢t is continuous and bounded in the vicinity of t=0 then the initial condition f⌃⁢0+=1, eq. (22), implies

for all 0<μ<1 and 0<α<1. [404.2.2.4] This is readily seen from bounding the integral in eq. (8) using the assumed continuity and boundedness of f⌃. [404.2.2.5] For 0<μ<1 and 0<α<1 equation (33) yields the result

independent of μ.

[404.2.3.1] Using equation (18) the susceptibility corresponding to the composite fractional relaxation equation is found as

for all 0<μ<1. [404.2.3.2] For τ1=τ2 this susceptibility function shows a broadened and asymmetric relaxation peak in the imaginary part. [404.2.3.3] Its asymmetrically broadened relaxation [page 405, §0]    peak resembles that of the Cole-Davidson [41] or Kohlrausch functions (see [42] for the Kohlrausch susceptibility).

[405.1.1.1] For composite fractional relaxation of type μ=1 one finds

and

[405.1.1.2] Composite fractional relaxation of type μ=1 was discussed in [35] in connection with the Basset force on a sphere moving under gravity in a viscous fluid. [405.1.1.3] In the following only the case 0<μ<1 is considered for fitting to experimental data.