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Preprint

We consider a fractional version of the classical nonlinear birth process of which the Yule--Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan--Caputo fractional derivative. We derive the probability distribution of the number \(\mathcal{N}_{\nu}(t)\) of individuals at an arbitrary time \(t\). We also present an interesting representation for the number of individuals at time \(t\), in the form of the subordination relation \(\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))\), where \(\mathcal{N}(t)\) is the classical generalized birth process and \(T_{2\nu}(t)\) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.

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Luisa Beghin, Enzo Orsingher (2004)

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Luisa Beghin, Enzo Orsingher (2011)

http://arxiv.org/licenses/nonexclusive-distrib/1.0/