Generalized Fractional Nonlinear Birth Processes

Mohsen Alipour (), Luisa Beghin () and Davood Rostamy ()
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Mohsen Alipour: Babol University of Technology
Luisa Beghin: Sapienza University of Rome
Davood Rostamy: Imam Khomeini International University

Methodology and Computing in Applied Probability, 2015, vol. 17, issue 3, 525-540

Abstract: Abstract We consider here generalized fractional versions of the difference-differential equation governing the classical nonlinear birth process. Orsingher and Polito (Bernoulli 16(3):858–881, 2010) defined a fractional birth process by replacing, in its governing equation, the first order time derivative with the Caputo fractional derivative of order υ ∈ (0, 1]. We study here a further generalization, obtained by adding in the equation some extra terms; as we shall see, this makes the expression of its solution much more complicated. Moreover we consider also the case υ ∈ (1, +∞ ), as well as υ ∈ (0, 1], using correspondingly two different definitions of fractional derivative: we apply the fractional Caputo derivative and the right-sided fractional Riemann–Liouville derivative on ℝ+, for υ ∈ (0, 1] and υ ∈ (1, +∞ ), respectively. For the two cases, we obtain the exact solutions and prove that they coincide with the distribution of some subordinated stochastic processes, whose random time argument is represented by a stable subordinator (for υ ∈ (1, +∞ )) or its inverse (for υ ∈ (0, 1]).

Keywords: Generalized fractional birth process; Fractional Caputo derivative; Fractional Riemann–Liouville derivative; Mittag–Leffler functions; Stable subordinator; 60G52; 34A08; 33E12; 26A33 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11009-013-9369-0

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