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Fractional relaxation with time-varying coefficient. (English) Zbl 1305.26018

Summary: From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

MSC:

26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
34A08 Fractional ordinary differential equations
76A10 Viscoelastic fluids
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