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General fractional calculus, evolution equations, and renewal processes. (English) Zbl 1250.26006

The author develops and studies a kind of fractional calculus connected certain Caputo type generalized fractional derivative, which involved a convolution integral. The main results presented by the author are based in the theory of complete Berstain functions.
The main goal of this interesting paper have been to study a Cauchy type problem for a linear parabolic equation involving the the mentioned generalized fractional derivative. Such stdy leads to an analytics description of general renewal processes constructed by M. M. Meerschaert, E. Nane and P. Vellaisamy [Electron. J. Probab. 16, 1600–1620 (2011; Zbl 1245.60084)] in terms of the random time change in the classical Poisson process determined by an inverse subordinator.

MSC:

26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
60K05 Renewal theory

Citations:

Zbl 1245.60084
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References:

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