[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 and type into the infinitesimal generator one arrives at the composite fractional relaxation equation in the form

(32) |

with two relaxation times and initial condition as before.

[404.2.2.1] A first advantage of the replacement over the replacement emerges when eq. (32) is Laplace transformed. [404.2.2.2] Using eq. (10) one finds

(33) |

[404.2.2.3] If the normalized relaxation function is continuous and bounded in the vicinity of then the initial condition , eq. (22), implies

(34) |

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

(35) |

independent of .

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

(36) |

for all . [404.2.3.2] For 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 one finds

(37) |

and

(38) |

[405.1.1.2] Composite fractional relaxation of type 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 is considered for fitting to experimental data.