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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 \alpha and type \mu into the infinitesimal generator \tilde{A}_{\alpha} one arrives at the composite fractional relaxation equation in the form

\tau _{1}\frac{\mathrm{d}}{\mathrm{d}t}\hat{f}(t)+\tau _{2}^{\alpha}D^{{\alpha,\mu}}_{{0+}}\hat{f}(t)+\hat{f}(t)=0 (32)

with two relaxation times 0<\tau _{1},\tau _{2}<\infty and initial condition \hat{f}(0+)=1 as before.

[404.2.2.1] A first advantage of the replacement T_{1}(t)\to\tilde{T}_{\alpha}(t) over the replacement T_{1}(t)\to T_{\alpha}(t) emerges when eq. (32) is Laplace transformed. [404.2.2.2] Using eq. (10) one finds

\hat{f}(u)=\frac{\tau _{1}\hat{f}(0+)+\tau _{2}^{\alpha}u^{{\mu(\alpha-1)}}(I^{{(1-\mu)(1-\alpha)}}_{{0+}}\hat{f})(0+)}{1+(\tau _{2}u)^{\alpha}+\tau _{1}u}. (33)

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

(I^{{(1-\mu)(1-\alpha)}}_{{0+}}\hat{f})(0+)=0 (34)

for all 0<\mu<1 and 0<\alpha<1. [404.2.2.4] This is readily seen from bounding the integral in eq. (8) using the assumed continuity and boundedness of \hat{f}. [404.2.2.5] For 0<\mu<1 and 0<\alpha<1 equation (33) yields the result

\hat{f}(u)=\frac{\tau _{1}}{1+(\tau _{2}u)^{\alpha}+\tau _{1}u} (35)

independent of \mu.

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

\hat{\chi}(u)=\frac{1+(\tau _{2}u)^{\alpha}}{1+(\tau _{2}u)^{\alpha}+\tau _{1}u} (36)

for all 0<\mu<1. [404.2.3.2] For \tau _{1}=\tau _{2} 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 \mu=1 one finds

\hat{f}(u)=\frac{\tau _{1}+\tau _{2}^{\alpha}u^{{\alpha-1}}}{1+(\tau _{2}u)^{\alpha}+\tau _{1}u} (37)


\hat{\chi}(u)=\frac{1}{1+(\tau _{2}u)^{\alpha}+\tau _{1}u}. (38)

[405.1.1.2] Composite fractional relaxation of type \mu=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<\mu<1 is considered for fitting to experimental data.